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Solid Modeling Using Implicit Solid Elements


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SOLID MODELING USING IMPLICIT SOLID ELEMENTS By JONGHO LEE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2003

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Copyright 2003 by Jongho Lee

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To my lovely family

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ACKNOWLEDGMENTS I would like to express my sincere gratitude to my advisor and the chairman of my supervisory committee, Dr. Ashok V. Kumar, for his guidance, encouragement and patience throughout my study. I feel myself very fortunate not only to have benefited from his academic guidance but also to have enjoyed his invaluable friendship. Without his assistance this study would never have been completed. I am also grateful to my committee members, Dr. John C. Ziegert, Dr. John K. Schueller, Dr. Carl C. Crane and Dr. Baba C. Vemuri, for their advice, comments and patience in reviewing this dissertation. I would also like to thank my parents, brothers, my wife, Duyoung Kim, and my lovely two daughters, Jasmine and Anna, for their valuable love and continuous support during my study. iv

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TABLE OF CONTENTS Page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES...........................................................................................................viii LIST OF FIGURES...........................................................................................................ix ABSTRACT.....................................................................................................................xiii CHAPTER 1 INTRODUCTION........................................................................................................1 1.1 Overview.................................................................................................................1 1.2 Goal and Objectives................................................................................................4 1.3 Outline....................................................................................................................4 2 SOLID REPRESENTATION SCHEMES...................................................................6 2.1 Geometric Modeling...............................................................................................6 2.2 Constructive Solid Geometry (CSG)......................................................................7 2.3 Boundary Representation (B-Rep).......................................................................11 2.4 Hybrid System......................................................................................................13 2.5 Sweep Features.....................................................................................................14 2.6 Implicit Surfaces or Level Set Surfaces................................................................15 3 IMPLICIT SOLID ELEMENTS................................................................................17 3.1 Definition of Implicit Solid Elements...................................................................17 3.2 Definition of the Solid within an Implicit Solid Element.....................................20 3.3 3D Hexahedral Solid Elements.............................................................................21 3.4 Defining and Editing Primitives...........................................................................23 3.5 Defining Complex Geometries using Implicit Solid Elements............................25 3.6 Heterogeneous Solid Model Capability................................................................26 4 TWO DIMENSIONAL PRIMITIVES.......................................................................28 4.1 2D Primitives by 9-Node Quadrilateral Elements................................................28 4.2 Mapping in 2D Solid Elements.............................................................................32 v

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5 CONSTANT CROSS-SECTION SWEEP.................................................................36 5.1 Extrude Elements..................................................................................................37 5.2 Revolve Elements.................................................................................................40 5.3 Sweep Elements....................................................................................................42 6 VARIABLE CROSS SECTION SWEEP..................................................................46 6.1 Straight Blend Element.........................................................................................47 6.2 Smooth Blend Element.........................................................................................55 6.3 Sweep Blend.........................................................................................................67 7 CSG REPRESENTATION USING IMPLICIT ELEMENTS...................................69 7.1 Surface Normal Vectors.....................................................................................69 7.2 Set membership classification within an element...............................................74 7.3 Constructive Solid Geometry using Implicit Solid Elements...............................77 8 VOLUME OF THE SOLID MODEL........................................................................88 8.1 Linear Approximate Step Function......................................................................88 8.2 Constructing the Step Function for the CSG Solid...............................................91 8.3 Computing the volume of the solid......................................................................93 9 ALGORITHM FOR GRAPHICAL DISPLAY........................................................101 9.1 Overview of Computer Graphics........................................................................101 9.2 2D Primitive Display..........................................................................................102 9.3 2D Boolean Result Display................................................................................104 9.4 3D Primitive Display..........................................................................................106 9.5 3D Boolean Result Display................................................................................108 9.6 Discussions and Suggestion for improvement....................................................116 10 CONCLUSION AND DISCUSSION......................................................................118 10.1 Conclusion........................................................................................................118 10.2 Future Work......................................................................................................119 APPENDIX SCENEGRAPH AND CLASS STRUCTURE................................................................120 A.1 SceneGraph Structure for Java3D.....................................................................120 A.2 SDModeler Classes...........................................................................................121 A.3 SDModeler Class Structure..............................................................................122 vi

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LIST OF REFERENCES.................................................................................................125 BIOGRAPHICAL SKETCH...........................................................................................127 vii

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LIST OF TABLES Table page 3-1. Basis functions for the 9 node quadrilateral element.................................................19 3-2. Hexahedral 8-node basis functions.............................................................................22 3-3. Hexahedral 18-node basis functions...........................................................................22 4-1. Samples of 2D primitive design.................................................................................30 4-2. Analytical solutions for the 4 node quadratic element mapping (parallelogram)......34 4-3. Analytical solutions for the 4 node quadratic element mapping (non-parallelogram)35 7-1. Surface normal vectors on the boundary of the element............................................73 8-1. Simple examples for Boolean operation between approximate step functions where f 1 and f 2 both represent squares....................................................................................90 8-2. Function operator denoted by B where h A and h B are any arbitrary step functions....93 8-3. Volume integration of Figure 8-6 by changing subdivision number where exact volume is 1.589 (Order of integration = 3, =0.001).............................................98 8-4. Volume integration of Figure 8-6 by changing the order of gauss integration where exact volume is 1.589 (Subdivision # = 4, =0.001).............................................99 8-5. Volume integration of Figure 8-6 by changing in the linear step function where exact volume is 1.589 (Order of integration = 3, Subdivision # = 4)......................99 8-6. Examples for computing the volume of the solid.......................................................99 viii

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LIST OF FIGURES Figure page 2-1. Example of CSG binary tree structure..........................................................................8 2-2. Comparison between the ordinary Boolean and regularized Boolean operation.........8 2-3. Regularized Boolean treatment in overlapped case......................................................9 2-4. Examples of neighborhood models on the vertices in 2D..........................................10 2-5. Mathematical description of the primitives by half-spaces A) Half-space by x-y plane B) Block by half spaces C) Cylinder by half spaces......................................11 2-6. Basic concept of B-Rep model...................................................................................12 2-7. Example of typical B-Rep data structure....................................................................13 3-1. Quadrilateral 9-node element in two different coordinate systems A) Parametric space (r, s coordinates) B) Real space (x, y : Global coordinates)..........................18 3-2. Density distribution within a 9 node quadratic element A) Density fringes B) Density contours C) Density Plot..........................................................................................20 3-3. Solids defined using 3D hexahedral elements A) 8 Node element B) 18 Node element.....................................................................................................................23 3-4. Editing the face represented using 2D element..........................................................24 3-5. Three ways of editing solids represented using 3D element......................................25 3-6. Implicit representation of planar face using shape density function A) Density Grid B) Contours of density function...............................................................................25 3-7. 3D Composition Destribution A) 8-node element B) 18-node element.....................27 4-1. 2D primitive examples A) Rectangle B) Circle C) Ellipse........................................28 4-2. Constructing a 2D primitive by the part of an existing shape....................................31 4-3. 2D primitives and its nodal density distribution A) Quarter circle B) Semi circle C) Triangle D) Wedge shape.........................................................................................31 ix

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5-1. 2D primitive element and the corresponding extrude element...................................37 5-2. Solid created by extruding ellipse..............................................................................39 5-3. Extrusion of profile defined using multiple elements................................................40 5-4.. Cylindrical coordinate system for revolving.............................................................40 5-5. Mapping from cylindrical to Cartesian coordinates...................................................41 5-6. Examples for revolved solids.....................................................................................42 5-7. Relationship between the global and local coordinate systems..................................43 5-8. Examples of solids created using sweep elements.....................................................45 6-1. Two neighboring straight blend elements having quadrateral 9 node elements as 2D profile.......................................................................................................................49 6-2. A straight blend element having quadratic 9 node elements as 2D profiles...............50 6-3. Cross section at a certain t in the straight blend element..........................................52 6-4. Examples of straight blend using blend elements......................................................55 6-5. Two neighboring smooth blend elements having quadratic 9 node elements as 2D profile.......................................................................................................................57 6-6. A smooth blend element having quadratic 9 node elements as 2D profile................60 6-7. Cross section at a certain t* in smooth blend elements..............................................62 6-8. Examples of Smooth blend using blend elements......................................................67 7-1. Flow chart for set-membership classification in an individual solid element............76 7-2. CSG tree data structure...............................................................................................77 7-3. 2D Boolean operation in CSG....................................................................................78 7-4. 3D Constructive Solid Geometry Tree.......................................................................79 7-5. surface normal vectors and corresponding triangles..................................................82 7-6. Constructing a triangle from one normal vector.........................................................82 7-7. Constructing two triangles from two normal vectors.................................................83 7-8. Constructing three triangles from three normal vectors.............................................84 x

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7-9. Neighborhood model using graphical method............................................................86 8-1. Linear approximate step function graph.....................................................................88 8-2. Transform from the implicit surface function to its linear approximate step function89 8-3. Implicit surfaces defining a solid within an element in the parametric space A) 2D solid element B) 3D solid element...........................................................................91 8-4. Labeling CSG tree nodes for constructing the approximate step-function................92 8-5. Subdivision of the 2D element and subdividing method A) Quadtree subdivision in a 2D element B) An example of quadtree representation structure C) An example of octree representation structure.................................................................................95 8-6. A simple 2D example used for comparing the numerical integration results by changing some parameters such as subdivision number, order of integration and ...............................................................................................................................98 9-1. Graphics of a curve...................................................................................................101 9-2. Graphics of a sphere and its triangulation................................................................102 9-3. Ray methods for finding points on the boundary of the solid in 2D........................103 9-4. Computing the points on the boundary of the ellipse...............................................104 9-5. Union of two squares in 2D......................................................................................105 9-6. Bounding box for a 2D solid element.......................................................................106 9-7. Computing the points on the boundary of the solid A) 2D profile to be swept B) Points on the extrude C) Points on the revolve D) Points on the sweep................107 9-8. Finding the points on the boundary of the varying cross section sweep..................108 9-9. 3D CSG tree structure...............................................................................................110 9-10. All triangleArrays composing the base and dependent solid.................................111 9-11. Bounding cylinder method between two elements.................................................112 9-12. All triangleArrays relevant to intersected elements...............................................112 9-13. Triangle-Triangle intersection A) Plane-Edge intersection B) Edge-Edge intersection.............................................................................................................113 9-14. TriangleArrays having actually intersected triangles.............................................113 xi

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9-15. Mapping the triangle in space onto the local u-v plane..........................................114 9-16. Subdivided triangles after Delaunay triangulation.................................................115 9-17. Flow chart for constructing the graphics for the 3D Boolean result......................116 A-1. SceneGraph structure for the solid modeler using implicit solid elements.............120 A-2. Class structure for the solid modeler using implicit solid elements........................122 A-3. ShapeDensity class structure...................................................................................123 A-4. InteractiveCommand class structure........................................................................123 A-5. CommandList class structure...................................................................................124 A-6. PositionConstraint class structure............................................................................124 xii

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SOLID MODELING USING IMPLICIT SOLID ELEMENTS By Jongho Lee August 2003 Chair: Ashok V. Kumar Major Department: Mechanical and Aerospace Engineering A solid modeling technique using implicit surface representation is presented where the implicit surface function (referred to here as density function) is defined by piece-wise interpolation within quadrilateral elements in 2D and hexahedral elements in 3D. Within each element the density function is defined as a parametric function. The solid is defined as the region within the element where the density is greater than a threshold or level set value. The boundary of the solid is therefore a contour of the density function along which the density is equal to the level set value or the boundary of the element where the density is greater than the level set value. Simple shapes or primitives can often be defined using a single implicit solid element while more complex geometry can be represented either using a grid/mesh of elements or by Boolean combination of simple primitives. The geometry of the primitives can be edited or modified by changing the density values at the nodes of the elements, by changing the size of the element as well as by changing the level set value. In this study, hexahedral xiii

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elements are presented along with basis functions that can be used to represent primitive solids created by sweeping 2D geometry along trajectories that can be lines, arcs or arbitrary parametric curves. In traditional solid modeling software based on the B-Rep (boundary representation) approach, solid primitives are often created by sweep operations. The elements described in this study enable the representation of such solids using implicit surfaces. This primitive representation scheme is axes independent because the implicit function representing the solid is defined as a parametric function within quadrilateral or hexahedral elements while general implicit surface functions are axes dependent. In general it is easier to determine whether a given point is inside, outside or on the boundary of a solid if the boundary is represented using implicit equations. Since it is necessary to repeatedly classify points in this manner for many graphical display and volumetric property evaluation algorithms implicit representation is particularly suited for representing primitives in a CSG tree. For the implicit solid element representation, it is also necessary to first map from global coordinates to parametric coordinates to perform this classification. This mapping is shown to be easy to perform if the elements have parallel edges and faces. The volume of the solid can be computed by defining the linear approximate step function, which has a unit value almost everywhere in the interior of the solid and zero in exterior of the solid. This linear approximate step function can be numerically integrated to compute the volume. xiv

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CHAPTER 1 INTRODUCTION 1.1 Overview Solid modeling is the fundamental technology on which CAD/CAM/CAE (Computer Aided Design/Manufacturing/Engineering) systems have been built. Solid modeling technique (sometimes called volumetric modeling) has been used for many applications including visualization, geometry design, assembly verification, engineering analysis and generating NC machining codes. The use of solid models for design and manufacturing is becoming more widespread with the increasing availability of computer technology. Solid models enable one to construct unambiguous models of three-dimensional objects. Such unambiguous models are necessary to construct algorithms for automatically computing volumetric properties such as surface area and volume of the solid. Algorithms for automatic generation finite element mesh for engineering analysis also depend on the ability to construct precise, unambiguous models of solids. Algorithms have also been developed to generate tool paths for machining given the solid model of the part to be machined. More recently developed fabrication and prototyping technology called the layered-manufacturing technique (also known as rapid prototyping or the solid freeform fabrication technique) also need solid models of the part to be constructed. The most popular schemes in current solid modeling are Boundary Representation (or B-Rep) and Constructive Solid Geometry (CSG). In B-Rep, a solid is represented by 1

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2 the boundary information of the solid where vertices, edges and faces (geometry information) are saved together with the information on how they are connected to build the boundary of the solid (topology information). The geometry of the boundaries are represented using parametric equations of the form X(u) or X(u,v) where X is a position vector of a point on the curve or surface. In CSG representation, Boolean operations applied on primitives that are simple shapes such as a block, cylinder, sphere or cone are saved in the binary tree data structure where primitives are defined by Boolean combination of half-spaces. Half spaces are implicit equations of curves or surfaces that are represented as f(x,y) 0 (curves) or f(x,y,z) 0 (surfaces). Current commercial solid modeling systems typically use a hybrid system where the solid is constructed by the Boolean combination of primitives that are represented using B-Rep models. The B-Rep model of the resultant solid is automatically constructed from the CSG tree using algorithms known as Boundary evaluation. In almost all current solid modeling systems, a feature based approach is provided as the user interface for creating solids where primitives are constructed by sweeping 2D profiles along a trajectory. Such sweep features for creating 3D primitives are popular because it enables the user to create primitives interactively. B-Rep solid models use parametric equations for geometric components such as lines, curves and surfaces. Creating B-Rep primitives by sweep operations is simple and straight forward because it is easy to define the equations of surfaces created by sweeping a curve along a trajectory in the parametric form. However, it is difficult to construct such sweep primitives by a Boolean combination of half-spaces as is done in traditional CSG systems.

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3 Solid modeling systems based on B-Rep models is very difficult and expensive to implement. The main reason for the difficulty is that the B-Rep model for the resultant geometry has to be constructed automatically from B-Rep models of the primitives using the information about the Boolean operations used to combine them. For each regularized Boolean operation, the boundary evaluator algorithm, detects intersections between participating solids, computes intersection geometries (intersection points and curves as well as sub-divided faces) and classifies them according to whether the geometry is in, on or outside the final solid defined by the Boolean operation. As mentioned earlier, B-Rep models need topology information as well as geometric information. Constructing connectivity (topology) between geometric components automatically is very difficult to be performed for complex geometries. And it is also very expensive to classify a given point, curve or face as being inside, outside or on the solid. The procedure for such classification is called set-membership classification. In traditional CSG models, the topology of the solid is not explicitly represented. Therefore, there is no need for Boundary evaluation algorithms. An algorithm is needed to generate the graphics of the resultant solid using the information available in the CSG tree. Even though in general the graphics image is difficult to generate for implicit curves and surfaces, the implementation of a CSG solid modeling system is easier because there is no need for Boundary evaluation algorithms. However, in order for such a modeling system to be useful it is necessary that a feature based modeling interface can be constructed. No attempt has been made to construct sweep features using implicit curves and surfaces since all current solid modeling systems use B-Rep models. Therefore one

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4 of the motivations of this study is to develop a feature based solid modeling system where all the boundaries of the solid are represented using implicit equations. 1.2 Goal and Objectives The goal of this research is to study the feasibility, advantages, and disadvantages of building a feature based solid modeling system that uses only implicit equations for curves and surfaces as it is done in traditional CSG solid modeling systems. Such solid models in this thesis are referred as implicit solid models. In order to develop such a system, implicit solid elements that are quadrilateral and hexahedral elements used to represent implicit curves and surfaces respectively have been used. The main objectives of the thesis are listed below. 1. Develop quadrilateral implicit solid elements to represents simple 2D primitives such as circles, ellipses, rectangles and triangles. 2. Develop hexahedral implicit solid elements to represent solid primitives obtained by extruding, revolving or sweeping 2D primitives defined using quadrilateral implicit solid elements. 3. Develop hexahedral implicit solid elements to represent a solid primitive obtained by blending between two or more different 2D primitives 4. Construct algorithms for displaying solids created by Boolean combinations of primitives represented using implicit solid elements. 5. Construct algorithms for automatically computing volumetric properties. 1.3 Outline In Chapter 2, current popular solid modeling techniques are introduced to give an overview for the reader. In Chapter 3, the definition of implicit solid elements is described and in Chapter 4, simple 2D primitives using 2D implicit solid elements are introduced. In Chapters 5 and 6, constant cross-section sweep elements and variable cross-section sweep elements are described respectively. Algorithms for set membership classification at a given point with respect to a primitive and a solid represented by a CSG tree structures are described in Chapter 7. In Chapter 8, the method to compute the

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5 volume of the solid is presented and in Chapter 9, issues associated with generating the computer graphics of the solid are described. Finally, Chapter 10 provides conclusions and discussions.

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CHAPTER 2 SOLID REPRESENTATION SCHEMES 2.1 Geometric Modeling Geometric Modeling systems are used in the design process for visualization, assembly verification, engineering analysis, generating NC machining codes, etc. It often eliminates the need for making many physical prototypes to test design concepts. There are three different types of geometric modeling systems: wireframe modeling, surface modeling and solid modeling. Wireframe Modeling is the earliest method, where a shape is represented by its characteristic lines and points. Although it has some advantages such as simple user inputs and ease of implementation, the following two disadvantages make this representation unpopular. First, the shapes represented only by lines and points are very ambiguous. Second, there is no information about the inside and outside boundary surfaces of the object. Therefore it cannot be used for mass property computation, tool path generation or finite element analysis. Surface Modeling has surface information in addition to the wireframe model information. Therefore, the tool path for the NC machines can be generated for the surfaces. However, since it also does not have information to distinguish between the inside and the outside of the object, Boolean operations cannot be used to combine such models and it is difficult or impossible to construct other algorithms for automatically manipulating such models as an example when generating a mesh. 6

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7 Solid Modeling was introduced to overcome some of the limitations of the other schemes mentioned above. Solid modeling techniques began to develop in the late 1960s and early 1970s. The fundamental concepts and definitions of solid modeling are well introduced in many books [Hoffmann, 1987, Zeid, 1991., Mortenson, 1997., Lee, 1999]. The most popular schemes in current solid modeling are Boundary Representation (or B-Rep) and Constructive Solid Geometry (CSG). In B-Rep, a solid is represented by the boundary information of the solid where vertices, edges, and faces are saved together with the information on how they are connected to build the boundary of the solid. In CSG representation, Boolean operations applied on primitives are saved in the binary tree data structure. These two representation schemes are described in the following section. 2.2 Constructive Solid Geometry (CSG) The constructive solid geometry representation technique defines complex solids as Boolean combinations of simpler solids called primitives. A Boolean operation is one of the best methods to create complex solids. It dramatically increases the repertoires of the shapes to be modeled. A binary tree data structure is used to save the Boolean operations between primitives. The following is an example of CSG tree data structure, where the final solid is generated by the union of the primitive A and B, followed by the difference of the primitive C. In this tree structure, leaf nodes (or primitive nodes) represent primitive solids and branch nodes have Boolean types such as union, difference, and intersect. Boolean operations can be explained by set theory. However, the set theory applied here is different from the original set theory. A modified Boolean operation has been defined for solid modeling which is referred to as Regularized Boolean operation, where the Boolean combination results between primitives are closed and dimensionally homogeneous.

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8 C U A B Figure 2-1. Example of CSG binary tree structure The following example shows how the regularized Boolean operation is different from the original Boolean operation. Regularized Boolean operations are denoted by * and respectively while the ordinary Boolean operations are denoted by , and Union Intersection Difference * B A B A Figure 2-2. Comparison between the ordinary Boolean and regularized Boolean operation As shown in Figure 2-2, problems always occur when the boundaries of the primitives overlap. These special cases can be treated to create a regularized result by comparing the directions of both boundary normal vectors. For example, in the regularized intersection, if both boundary normal vectors have the same direction, the

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9 boundary is also the boundary of the regularized Boolean combination. On the contrary, if they have opposite directions, it should be removed to construct the regularized Boolean combination. The following shows all-special cases, where n A and n B are the boundary normal vectors of the primitive A and B at the overlapped part respectively. ON IN ON OUT OUT ON B nB A nA B A nB nA Figure 2-3. Regularized Boolean treatment in overlapped case The method using normal vectors for the regularized Boolean combination in solid modeling cannot be used when multiple normal vectors are involved. For instance vertices in 2D or 3D and edges in 3D cannot be simply classified as in, out or on the solid because those have multiple normal vectors. This problem can be solved by using a neighborhood model where points close to the given point (or in its neighborhood) are examined. Figure 2-4 shows an example of a neighborhood model where a small disk symbolizes the neighborhood, and the shaded area indicates points inside the solid. The neighborhood model at a point for the Boolean result is constructed by applying the Boolean operation to the neighborhood models at the point for the two participating solids.

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10 Figure 2-4. Examples of neighborhood models on the vertices in 2D A B ABBAOnOnAUBAUB There are three important subsets defined by a solid, where the solid is considered a set of points. Those are the set of interior points, points on the boundary of the solid, and all points outside the solid. Set-membership classification of a point involves assigning the point to one of these sets. It is an essential step to do a Boolean operation. Primitives are simple shapes such as a block, cylinder, sphere or cone, where the mathematical descriptions of the primitives are defined by the intersection of a set of curved or planar half-spaces. A half space divides the space into two regions, for instance, an infinite x-y plane divides the space into the region of z>0 and the region of z<0 as shown in Figure 2-5A. Figure 2-5B and Figure 2-5C show how to define the primitive block and cylinder by half-spaces respectively. Where the block is represented by the intersection of six planar half-spaces and the cylinder is represented by the intersection of a cylindrical half-space and two planar half-spaces.

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11 a x byez azbz z z>0 z y z<0 x (A) (B) (C) Figure 2-5. Mathematical description of the primitives by half-spaces A) Half-space by x-y plane B) Block by half spaces C) Cylinder by half spaces Arrows indicate the direction of the material. The detailed mathematical descriptions can be represented as follows: Block : bxa dyc fze Cylinder : 0, 222ryx bza This kind of description is good for set membership classification because the sign of the implicit function representing the half space can be used to classify the point. CSG representation does not include topology information and therefore additional computation is necessary to determine the connectivity between boundary entities. Such connectivity information is used in many applications such as mesh generation and tool path generation for machining. 2.3 Boundary Representation (B-Rep) The B-Rep model of a solid consists of parametric equations of the vertices, edges and faces of the solid together with the connectivity information between them. The geometry information in a B-Rep model is composed of surface equations, curve

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12 equations, and point coordinates. The connectivity between the geometric entities is the topology information which is the interrelationship among faces, edges and vertices. It is based on the idea that the boundary of a solid consists of faces (surfaces) bounded by edges (curves), which in turn are bounded, by vertices (points). In B-Rep, a face is a subset or limited region of some more extensive surface. For example, planar faces are subsets of infinite planes that are bounded by edges, while curved faces are represented as parametric surfaces bounded by edges. Thus, these equations are saved in the B-Rep data structure. The topology information is represented as relations between faces, edges, and vertices. In Figure 2-6, the face F 1 is composed of four edges (E 1 E 2 E 4 E 3 ) and each edge is represented by two vertices. E1 V1 V2 F3 F1 F1 E2 E4 F2 V4 V3 E3 Figure 2-6. Basic concept of B-Rep model An array of faces, edges, and vertices can be used to represent this kind of simple polyhedral model. Figure 2-7 shows the typical data structure [Spatial Technology Inc., 1995] used to represent B-Rep models which consists of shells, faces, loops, half edges(or coedges), and vertexes. Figure 2-7 shows classes used to represent the solid model. A solid is denoted as a BODY which consists of one or more LUMPs. A LUMP is a connected 3D region whose boundaries are represented using closed SHELLs. A closed SHELL consist of a list of faces forming a closed volume. As mentioned earlier, the geometry of a face is defined by a parametric surface equation and is bounded by LOOPs that define its external and internal boundaries. Loops are defined as a list of half-edges (COEDGEs), which are

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13 edges with a direction. Half edges are connected together to form loops such that the direction of the loop is counter clockwise for the external boundary and clockwise for internal boundaries when the face is viewed from the outside. Assigning directions to loops in this manner assists in determining whether a given point is inside, on, or outside the solid. If the solid has a lot of faces, this kind of set membership classification is difficult and it requires significant computation. Since every entity is interconnected with other entities, changing the topology is very complicated in the B-Rep model approach. F1 V1 E1 C5 C1 V2 L1 L2 Ex ) E1V1 V2 C1E1 L1C1 C2 C3 C4 F1L1 L2 S1F1 F2 .. F10 Lum p 1S1 next p revious nex t nex t POINT CURVE SURFACE VERTEX EDGE COEDGE LOOP FACE SHELL LUMP BODY Figure 2-7. Example of typical B-Rep data structure 2.4 Hybrid System Creating B-Rep models directly by defining each vertex, edge, and surface is extremely cumbersome. Therefore, hybrid systems were developed where the solid is represented procedurally using a CSG tree but the primitives are represented using B-Rep. The B-Rep model of the solid represented by the CSG tree is evaluated

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14 automatically using Boundary Evaluator algorithms [Requicha and Voelcker, 1985]. For each regularized Boolean operation, the boundary evaluator algorithm, detects intersections between participating solids, computes intersection geometries (intersection points and curves as well as sub-divided faces) and classifies them using set membership classification algorithms to determine whether the geometry is in, on, or outside the final solid defined by the Boolean operation. While set membership classification is easy when the solid is represented as a combination of half spaces (where the surfaces are implicit equations, f(x,y,z) = 0), it is more difficult to do for B-Reps. Furthermore, constructing the topology of the resultant solid automatically is also difficult. As a result it is very expensive to build robust and reliable software that can handle every special case. However, over the years many commercial systems have been developed that have very reliable boundary evaluator algorithms that are robust and can handle almost all special and degenerate cases including non-manifold topologies. To cope with the high cost of implementing these algorithms, many companies in the CAD industry buy geometric modeling kernels from other companies that have already implemented them [Spatial Technology Inc., 1995]. 2.5 Sweep Features A sweep feature is defined by sweeping a planar shape along an arbitrary space curve referred to as a sweep trajectory, where the cross section of the sweep solid can be either constant or varying. This is a very popular and useful method due to the fact that the solid represented by sweeping is simple to understand and execute. If the sweep trajectory is a straight line, it is called as translational sweep (or extruded solid). When the sweep trajectory is a circular arc, it is termed a rotational sweep (or revolved solid). When the sweep trajectory is an arbitrary parametric curve, it is referred to as a general

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15 sweep. Recent research related to sweeping includes three-dimensional object sweeping and sweep surface representation using coordinate transforms and blending [Martin and Stephenson, 1990, Choi and Lee, 1990]. 2.6 Implicit Surfaces or Level Set Surfaces Traditionally, implicit curves and surfaces are represented as f(x) = c where 2R x for planar curves and for surfaces. They are sometimes referred to as level sets or iso-curve/surface since the curve or surface corresponds to a constant value of the function f(x). If c = 0 then the curve or surface is called the zero set of f(x). An implicit surface divides space into regions where f(x) > 0 and f(x) < 0. Therefore, if the boundaries of the solid are represented implicitly as half-spaces one can use the sign of f(x) to perform set membership classification that is to determine whether a given point is inside, outside, or on the solid. Despite this advantage, implicit representation for curves and surfaces has traditionally been considered inferior to parametric representation. The reasons suggested for this include axis dependence of implicit curves, difficulty in tracing or generating graphics, as well as difficulty in fitting and manipulating freeform shapes. Considerable progress has been made to solve some of these problems due to which implicit curves and surfaces now have numerous applications in graphics and animation [Bloomenthal et al., 1997]. 3Rx To enable graphical display a variety of polygonization, tessellation, or tracing algorithms have been developed [Wyvill and Overveld, 1997., Lorensen and Cline, 1987]. Ray tracing algorithms have also been used for visualization of implicit surfaces [Glassener, 1989]. When the implicit function f(x) is a polynomial, the surface is called an algebraic surface. Most common primitive shapes such as sphere, ellipse, cone, and

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16 cylinders can be expressed as algebraic surfaces using quadratic polynomials. For more complex shapes, implicit algebraic surface patches (or A-splines) have been developed [Bajaj et al., 1995]. These patches can be used for C 1 and C 2 interpolations or approximations and also for interactive free-form modeling schemes. Implicit curves and surfaces have also been used for visualization problems such as shape reconstruction from unorganized data sets [Zhao et al., 2000] and dynamic fluid flow [Osher and Redkiw, 2003]. Recently, R-functions [Shapiro, 1998] have been used to define implicit solids. The sign of an R-function depends only on the sign of its arguments and not its magnitude. This property of R-functions can be used to construct implicit functions for representing solids created by a Boolean combination of implicit surfaces. Furthermore, the advantage using geometry represented using implicit curves and surfaces in solving boundary and initial value problems with time varying geometries and boundary conditions has also been illustrated [Shapiro and Tsukanov, 1999]. Implicit surfaces are often used in computer graphics for representing soft or deformable objects such as humans, animals, and amorphous blobs. Sometimes, an implicit surface can be represented by the distance relationship between the surface and a given basic structure such as point, line, and circle. This kind of method is called distance metrics or the skeleton method [Tigges and Wyvill, 1999].

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CHAPTER 3 IMPLICIT SOLID ELEMENTS 3.1 Definition of Implicit Solid Elements The implicit curves and surfaces are represented here as the level set of an implicit surface function (r,s) = b for planar faces or (r,s,t) = b for solids. The implicit surface function, has been referred to in this study as the shape density function (or simply density function) [Kumar and Gossard, 1996]. The density function has a value greater than a threshold or level set value, b inside the solid and less than b outside the solid. Unlike the traditional approach, where the implicit surface function is defined directly in terms of the x, y, z or Cartesian coordinates, the density function used here is a parametric function whose arguments are the parametric coordinates of an implicit solid element. This is illustrated using a two dimensional example in Figure 3-1 where the element has nine nodes and its geometry in the parametric space is shown in Figure 3-1A while its real geometry is shown in Figure 3-1B. In the parametric space the element is a square whereas in the real space the element can be a distorted quadrilateral. The mapping between the parametric coordinates and the real coordinates x, y and z is defined as, niii1niii1niii1x(r,s,t)xM(r,s,t), y(r,s,t)yM(r,s,t) and z(r,s,t)zM(r,s,t) (3-1) 17

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18 In Eq. 3-1. (x i y i z i ) are the coordinates of the node i, are mapping basis functions that define the mapping between the parametric space (r,s,t) and the real space (x,y,z) and n is the total number of nodes per element. (x,y,z) and n is the total number of nodes per element. iM(r,s,t) (A) (A) (B) (B) 5 = b 9 1 8 4 3 7 6 2 y x 9 1 8 3 7 4 6 5 2 s r Figure 3-1. Quadrilateral 9-node element in two different coordinate systems A) Parametric space (r, s coordinates) B) Real space (x, y : Global coordinates) Figure 3-1. Quadrilateral 9-node element in two different coordinate systems A) Parametric space (r, s coordinates) B) Real space (x, y : Global coordinates) The density functions are defined within the elements by specifying the values of the functions at the nodes of these elements. The value of the function within each element is obtained by interpolating the values at the nodes using appropriate basis functions or interpolation functions. A parametric interpolation scheme is used as shown below where parameters (r, s, t) vary between and 1. The density functions are defined within the elements by specifying the values of the functions at the nodes of these elements. The value of the function within each element is obtained by interpolating the values at the nodes using appropriate basis functions or interpolation functions. A parametric interpolation scheme is used as shown below where parameters (r, s, t) vary between and 1. niiisrNsr1),(),(niiisrNsr1),(),( for 2D elements (3-2) for 2D elements (3-2) niiitsrNtsr1),,(),,(niiitsrNtsr1),,(),,( for 3D elements (3-3) for 3D elements (3-3) Where, i is the density at the node i, N i is the interpolation basis function at the node i used to interpolate the density, and n is the total number of nodes in the element. Each element is a cube (or square in 2D) of side length equal to 2 in the parametric space since the parameters r, s and t vary from -1 to 1 within each element. If the mapping basis functions are identical to the density interpolation basis functions Where, iM(r,s,t)iM(r,s,t) iN(r,s,t)iN(r,s,t) i is the density at the node i, N i is the interpolation basis function at the node i used to interpolate the density, and n is the total number of nodes in the element. Each element is a cube (or square in 2D) of side length equal to 2 in the parametric space since the parameters r, s and t vary from -1 to 1 within each element. If the mapping basis functions are identical to the density interpolation basis functions

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19 then the element can be referred to as iso-parametric element. However, it is often beneficial to use simpler basis functions for the mapping. In this formulation, even though the boundary of the solid is represented in the implicit form (x,y,z) = b the density function itself is represented in a parametric form, (r, s, t) with a mapping defined between the parameters (r,s,t) and the global coordinates (x,y,z). Therefore, many of the advantages traditionally associated with parametric curves and surfaces are also applicable to the implicit solid element approach including axes independence and the ease in tracing. The parametric form of the density function enables the development of simple algorithms to polygonize the surface for graphical display. The detailed description for one way of graphical display used in this system is explained in Chapter 9. The following table shows the basis functions for the 9 node quadrilateral element where the numbering system is as shown in Figure 3-1. Table 3-1 Basis functions for the 9 node quadrilateral element N 1 = 0.25(r 2 +r)(s 2 +s) N 2 = 0.25(r 2 -r)(s 2 +s) N 3 = 0.25(r 2 -r)(s 2 -s) N 4 = 0.25(r 2 +r)(s 2 -s) N 5 = 0.5(1-r 2 )(s 2 +s) N 6 = 0.5(r 2 -r)(1-s 2 ) N 7 = 0.5(1-r 2 )(s 2 -s) N 8 =0.5(r 2 +r)(1-s 2 ) N 9 = (1-r 2 )(1-s 2 ) These shape functions are derived by Lagrange interpolation such that N i is, equal to 1 at node i and is zero at other nodes. For instance, N 1 = 0 at node 2 to 9 is equivalent to requiring that N 1 = 0 along edges r = 0, r = -1, s = 0 and s = -1. Therefore N 1 can be represented by N 1 = c(r 2 +r)(s 2 +s) where c is some constant that is determined from the condition N 1 = 1 at node 1. Since r = 1 and s =1 at node 1, yields

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20 1 = c(1 2 +1)(1 2 +1)= 4c Which yields c = 0.25. Similarly other shape functions at nodes 2 to 9 are derived as shown in Table 3-1. Figure 3-2 shows the density distribution within a 9-node element when the density at the corner nodes are set equal to 0, the density at mid-edge nodes are set to 0.5 and the density is 1 at the center node. (A) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 rs0.40.40.50.50.50.50.50.60.60.60.60.60.60.60.60.70.70.70.70.70.70.80.80.80.80.80.90.90.90.9 (B) (C) Figure 3-2 Density distribution within a 9 node quadratic element A) Density fringes B) Density contours C) Density Plot In Figure 3-2B each contour line corresponds to constant shape density values. Therefore the boundary of the solid in this system is one of the contours whose value is equal to the boundary density value ( b ). For instance if 0.5 is set to b the boundary of the solid would be a circular shape as highlighted in Figure 3-2A. 3.2 Definition of the Solid within an Implicit Solid Element In brief, the definition of the solid used here can be represented as b where is a density value defined at the isoparametric coordinates namely, 1 1 r and in 2D (and 11s 11 t in 3D). Therefore, the solid in this system may now be defined as the set of points S such that

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21 11,11),(,),(11srwheresrMXsrNniiibniiiXXS for 2D elements 11,11,11),,(,),,(11tsrwheretsrMXtsrNniiibniiiXXS for 3D elements (3-4) where, 2 R X for 2D elements and 3 R X for 3D elements. Since the boundary of the geometry is very critical in some cases, clear definition of the boundary of the solid is required in any solid modeling system. In this system there are two criterions defining the boundary of the solid. Therefore, the boundary of the solid represented by an element can be defined as the set of points B such that 11),(,),(11,11),(,),(1111sorrwheresrMXsrNANDsrwheresrMXsrNniiibniiiniiibniiiXXXXB for 2D elements 111),,(,),,(11,11,11),,(,),,(1111torsorrwheretsrMXtsrNANDtsrwheretsrMXtsrNniiibniiiniiibniiiXXXXB for 3D elements (3-5) 3.3 3D Hexahedral Solid Elements Three-dimensional primitives can be defined using hexahedral elements such as hexahedral 8-node or 18-node elements. Figure 3-3 shows these two 3D elements and the node numbering scheme used. The basis functions of a hexahedral 8-node element and the interpolation within each element are trilinear. Therefore, the higher order 18-node

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22 element is required to create cylindrical and conical shapes. The basis functions of the 8-node and 18-node hexahedral element are listed in the following tables. Table 3-2 Hexahedral 8-node basis functions N1 = 0.125(1+r)(1+s)(1+t) N2 = 0.125(1-r)(1+s)(1+t) N3 = 0.125(1-r)(1-s)(1+t) N4 = 0.125(1+r)(1-s)(1+t) N5 = 0.125(1+r)(1+s)(1-t) N6 = 0.125(1-r)(1+s)(1-t) N7 = 0.125(1-r)(1-s)(1-t) N8 = 0.125(1+r)(1-s)(1-t) Table 3-3 Hexahedral 18-node basis functions N1= 0.125(1+r)(1+s)(1+t)rs N2= -0.125(1-r)(1+s)(1+t)rs N3= 0.125(1-r)(1-s)(1+t)rs N4= -0.125(1+r)(1-s)(1+t)rs N5= 0.125(1+r)(1+s)(1-t)rs N6= -0.125(1-r)(1+s)(1-t)rs N7= 0.125(1-r)(1-s)(1-t)rs N8= -0.125(1+r)(1-s)(1-t)rs N9= 0.25(1-r 2 )(1+s)(1+t)s N10= -0.25(1-r)(1-s 2 )(1+t)r N11= -0.25(1-r 2 )(1-s)(1+t)s N12= 0.25(1+r)(1-s 2 )(1+t)r N13= 0.5(1-r 2 )(1-s 2 )(1+t) N14= 0.25(1-r 2 )(1+s)(1-t)s N15= -0.25(1-r)(1-s 2 )(1-t)r N16= -0.25(1-r 2 )(1-s)(1-t)s N17= 0.25(1+r)(1-s 2 )(1-t) N18= 0.5(1-r 2 )(1-s 2 )(1-t) Figure 3-3 shows two solids defined using these hexahedral elements with the boundary density value 5.0b In Figure 3-3A, an 8-node element is used with the following values of the nodal densities: 1 =0.4 2 =1.0 3 =0.2 4 = 1.0 5 = 0.4 6 = 0.9 7 = 0.3 8 = 1.0. Figure 3-3B shows a cylindrical solid with the elliptical cross-section created using the 18-node element. The density values at the nodes were set as follows to create this geometry: 1 = 2 = 3 = 4 = 5 = 6 = 7 = 8 =0.0, 9 = 10 = 11 = 12 = 14 = 15 = 16 = 17 =0.5, and 13 = 18 = 1. 1 2 3 4 5 6 8 s r t 9 10 11 12 13 14 15 16 17 18 t r 8 7 6 5 4 3 2 1 s

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23 (A) (A) (B) (B) Figure 3-3 Solids defined using 3D hexahedral elements A) 8 Node element B) 18 Node element Figure 3-3 Solids defined using 3D hexahedral elements A) 8 Node element B) 18 Node element In the following section, the method to define and edit primitives using implicit solid elements is described. In the following section, the method to define and edit primitives using implicit solid elements is described. 3.4 Defining and Editing Primitives 3.4 Defining and Editing Primitives The geometry defined within each element can be edited by modifying the density values at each node, the shape of the element. or level set value of density. This diversity in editing methods enables the creation of a variety of primitives using a single element. Figure 3-4 shows how the solid geometry is changed in response to the change in such parameters as density value at each node, nodal coordinates of the element, and boundary density value. In Figure 3-4A, the density value at the nodes of the 9-node element is set such that the density at all nodes is set to b = 0.5, except the central node where density is set equal to 1. The geometry is therefore identical to that of the element since every point within has density greater than b The geometry in Fig. 3-4B is obtained by changing the density values at nodes to The geometry defined within each element can be edited by modifying the density values at each node, the shape of the element. or level set value of density. This diversity in editing methods enables the creation of a variety of primitives using a single element. Figure 3-4 shows how the solid geometry is changed in response to the change in such parameters as density value at each node, nodal coordinates of the element, and boundary density value. In Figure 3-4A, the density value at the nodes of the 9-node element is set such that the density at all nodes is set to b = 0.5, except the central node where density is set equal to 1. The geometry is therefore identical to that of the element since every point within has density greater than b The geometry in Fig. 3-4B is obtained by changing the density values at nodes to 1 =0.2, 2 =0.3, 3 =0.4, 4 = 5 = 6 = 7 = 8 = 0.5 and 9 =1. Changing the shape of the element as shown in Figure 3-4C where the element nodes are moved to stretch the element into a rectangle can also modify the geometry.

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24 Note that the element shape should be a parallelogram in this system. This method of editing the geometry enables a variety of quadrilateral primitives, such as rectangles, trapezoids and parallelograms, to be constructed by deforming the element in Figure 3-4A. (A) (B) (C) (D) Figure 3-4 Editing the face represented using 2D element Yet another way to modify the geometry is by changing the level set value, b so that a different level set or contour of the density function becomes the boundary of the solid. This is illustrated in Figure 3-4, where the solid in Figure 3-4D is obtained from the solid in Figure 3-4C by changing the level set value from 0.5 to 0.6. Figure 3-5 shows the above three methods of editing for a 3D element. The solid in Figure 3-5B is obtained by just changing the density values of the solid in Figure 3-5A and it is further modified by changing the shape of the element to obtain the solid in Figure 3-5C. Finally, the solid in Figure 3-5D is obtained by modifying the level set value b

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25 Figure 3-5 Three ways of editing solids represented using 3D element (A) (B) (C) (D) 3.5 Defining Complex Geometries using Implicit Solid Elements Simple solid primitives may be defined using a single implicit solid element. Using a grid of elements as illustrated in Figure 3-6, it is possible to create more complex primitives. Figure 3-6A shows the two-dimensional mesh or grid used to represent the shape and Figure 3-6B shows the density values using gray scale such that white is = 1 and black is = 0. The boundary of the geometry is represented partly by the contours of the density function = 0.5 (which is highlighted) and partly by the boundaries of the mesh where 0.5. When the primitive is modeled using a mesh that has many elements as in Figure 3-6, the solid model is the union of the solids defined by the individual elements in the mesh. (A) (B) Figure 3-6 Implicit representation of planar face using shape density function A) Density Grid B) Contours of density function

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26 A variety of elements and basis functions can be defined and used to represent the solids in 2D and 3D. Higher degree polynomials can be used as basis functions for elements with a larger number of nodes per element. If the geometry to be represented is simple (e.g. rectangle, circle or ellipse) then it is often possible to define the required shape density function using just one element. Primitives thus defined using one or more elements can be combined using various Boolean operations to create a procedural definition of more complex solids, expressed as a CSG tree. Refer to Chapter 7 for the detailed description about CSG representation scheme used in this system. 3.6 Heterogeneous Solid Model Capability Conventional solid modeling systems have been developed based on the assumption that the solid is perfectly homogeneous. Thus they cannot represent internal properties or offer ways of representing internal behavior. In this solid representation using implicit solid elements, material composition can be defined within each element by interpolating nodal values of composition. Many biomechanical structures such as bones, shells, and muscular tissues are heterogeneous. Once heterogeneous objects are modeled as the solid models, they can be manufactured by recently developed manufacturing methods such as layered manufacturing technique, also known as rapid prototyping or solid freeform fabrication. The following examples illustrate this idea where material composition distributions are represented within solids by defining composition values at each node and using the same interpolation scheme used in defining geometries.

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27 (A) (B) Figure 3-7 3D Composition Destribution A) 8-node element B) 18-node element In this case, two materials termed A, B respectively are used for composing the heterogeneous objects. The A material is shown as red in Figure 3-7 and the material B is shown as blue. Let be the volume percentage of material A at a point. Then 1is the volume percentage of B at the same point. The nodal density values are larger than the boundary density value ( b ) for both examples in Figure 3-7. The material compositions (volume percentage of material A) at the nodes for creating Figure 3-7 were defined as follows: (A) 1 = 2 = 5 = 6 =1.0 3 = 4 = 7 = 8 =0.0 (B) 1 = 2 = 3 = 4 = 5 = 6 = 7 = 8 =0.0, 9 = 10 = 11 = 12 = 14 = 15 = 16 = 17 =0.5, and 13 = 18 = 1 (3-6)

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CHAPTER 4 TWO DIMENSIONAL PRIMITIVES As mentioned in the proceeding chapter, simple primitives are combined using Boolean operations to create more complex solids. In this chapter, 2D implicit elements used for constructing 2D primitives are presented. 4.1 2D Primitives by 9-Node Quadrilateral Elements A quadrilateral primitive can be represented directly using quadrilateral elements by setting the nodal values of density to be greater than or equal to the level set value. While it is possible to create a variety of implicit solid elements, the 9-node quadrilateral element for two-dimensional primitives such as quadrilaterals, circles and ellipses has been used primarily. A variety of elements and interpolation functions can be used for defining primitives however for brevity and consistency a 9-node element was chosen among lots of capable elements to create all primitives in this system. The following figures show examples of 2D primitives defined by the quadratic 9-node element. (A) (B) (C) Figure 4-1 2D primitive examples A) Rectangle B) Circle C) Ellipse 28

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29 Rectangular primitives as shown in Figure 4-1A can be defined by giving density values larger than the boundary density value ( b ) at each node. Then any rectangular shapes can be created by modifying the size of the element. A primitive circle can be defined using a single 9-node quadrilateral element. The boundary is defined as ),(sr = = 91),(iiisrN b = 0.5 (4-1) Expanding the above equation by substituting N i listed in Table 3-1, yields 0.5 = N 1 1 + N 2 2 + N 3 3 + N 4 4 + N 5 5 + N 6 6 + N 7 7 + N 8 8 + N 9 9 (4-2) = 9 + (0.5 6 .5 8 )r + (-0.5 5 +0.5 7 )s + (0.5 6 +0.5 8 9 )r 2 + (0.5 5 +0.5 7 9 )s 2 + (0.25 3 .25 2 -0.25 4 .25 1 )rs + (-0.25 2 +0.5 5 +0.25 4 +0.25 3 0.25 1 .5 7 )r 2 s + (-0.25 1 +0.25 2 +0.25 3 -0.25 4 0.5 6 +0.5 8 )r s 2 + ( 9 +0.25 1 -0.5 5 -0.5 6 + 0.25 4 +0.25 3 +0.25 2 -0.5 7 -0.5 8 )r 2 s 2 To be an equation of circle, the coefficients of r, s, rs, r 2 s, rs 2 and r 2 s 2 should be equal to zero and the coefficients of r 2 and s 2 must be same. In other words, the following conditions should be satisfied. (0.5 6 .5 8 ) = 0 (-0.5 5 +0.5 7 ) = 0 (0.25 3 .25 2 -0.25 4 .25 1 ) = 0 (-0.25 2 +0.5 5 +0.25 4 +0.25 3 0.25 1 .5 7 ) = 0 (-0.25 1 +0.25 2 +0.25 3 -0.25 4 0.5 6 +0.5 8 ) = 0 ( 9 +0.25 1 -0.5 5 -0.5 6 + 0.25 4 +0.25 3 +0.25 2 -0.5 7 -0.5 8 ) = 0 (0.5 6 +0.5 8 9 ) = (0.5 5 +0.5 7 9 ) (4-3) From the above equations, the following conditions are derived. 1 = 2 = 3 = 4 5 = 6 = 7 = 8

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30 9 =2 5 1 9 5 (4-4) This results in a circle of unit radius within the element in the parametric space whose equation is r 2 + s 2 = 1. The geometry in the real space will also be a circle if the shape of the element is square and diameter of the circle will be equal to the size of the square as shown in Figure 4-1B. If the element shape is a rectangle in real space, then the circle in the parametric space maps into an ellipse in the real space with the major and minor diameters equal to the length and height of the rectangle as shown in Figure 4-1C. Clearly there is no unique representation for any given primitive. In the following examples quadrilateral 9node elements are used for all primitives. Table 4-1: Samples of 2D primitive design b Element shape i (Nodal densities) Rectangle 0.5 Square 1 = 2 = 3 = 4 = 5 = 6 = 7 = 8 = 0.5 and 9 =1 Circle 0.5 Square 1 = 2 = 3 = 4 =0.0, 5 = 6 = 7 = 8 = 0.5 and 9 =1 Ellipse 0.5 Rectangle 1 = 2 = 3 = 4 =0.0, 5 = 6 = 7 = 8 = 0.5 and 9 =1 Figure 4-2 shows more 2D primitive examples created using the 9-node elements. The value of density at the node is displayed near each node. The density values at each node can be found using the same approach that was used for computing the nodal density for the circle. Alternatively, one can make use of the property of Lagrange interpolation that the interpolation obtained is unique. Therefore if the implicit equation of a curve is known then the value of the function evaluated at the nodal coordinates can be assigned as the nodal values.

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31 01 5.05 025.06 5.08 19 5.07 04 03 )1,1(1)1,5.0(5)1,0(2)5.0,0(6 )5.0,1(8)5.0,5.0(9 )0,5.0(7 )0,1(4 )0,0(3 Figure 4-2 Constructing a 2D primitive by the part of an existing shape Figure 4-2 shows how to compute nodal density values for a quarter circle primitive from the primitive circle where the density function used is as follows: 291221211),(),(srsrNsriii 5.05 01 5.04 875.09 17 5.03 02 375.08 875.06 01375.055.02875.06375.0875.09875.075.0413 (A) (B) 625.05 5.02 11125.06 25.09 03 125.07 5.04 625.08 25.05 5.02 01 75.06 5.09 1375.075.0425.08 (C ) (D) Figure 4-3. 2D primitives and its nodal density distribution A) Quarter circle B) Semi circle C) Triangle D) Wedge shape

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32 4.2 Mapping in 2D Solid Elements The 9-node quadrilateral element has bi-quadratic basis functions as shown in Table 3-1. The same basis functions could be used to define the mapping between the parametric space and the real space. However, it can be shown that the same mapping is obtained when the bilinear basis functions are used if the element is a parallelogram with straight edges in the real space and the mid nodes are at the center of each edge. In addition the center node should be located at the center of the element. Using the bilinear basis function for mapping simplifies inverse mapping from real space to the parametric space. If the biquadratic basis functions are used, the mapping equations are non-linear simultaneous equations having multiple solutions in general. The following equations show the 4-node mapping basis functions. 123411M1r1s,M1r1s4411M1r1s,M1r1s44 (4-5) A simple proof is given below to show that the mapping from the real space to the parametric space using biquadratic basis functions of 9 node quadrilateral elements is identical to the mapping using bilinear 4 node basis functions if the element has straight edges with mid nodes at mid points of each edge and the center node at the center of the element. The mapping equations using biquadratic basis functions are as follows 91),(iiisrMXX 922822722622522422322222122)s-)(1r-(1)s-r)(1(r21s)-)(sr-(121)s-r)(1-(r21 s))(sr-(121s)-r)(s(r41 s)-r)(s-(r41s)r)(s-(r41s)r)(s(r41XXXXXXXXX (4-6)

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33 where X i is the nodal coordinates at node i and the numbering scheme used corresponds to Figure 2-1. The condition that mid-edge nodes are at the mid points of each edge and the center node is at the centroid of the parallelogram can be stated as follows. 2215XXX 2326XXX 2437XXX 2148XXX 12394XXXXX 4 (4-7) Substituting Eq. 4-7 into Eq.4-6 and rearranging yields 41432191),(s)r)(1(141 s)r)(1(141s)r)(1(141s)r)(1(141),(iiiiiisrMsrMXXXXXXX (4-8) The above equation shows that the mapping for the 9 node element is identical to mapping for the 4 node element when the 9 node element is a parallelogram satisfying the conditions in Eq. 4-7. Parametric coordinates corresponding to a given global coordinates are unique for a 4-node quadrilateral element when it is a parallelogram. To make the calculation simple and fast, an analytical solution for the quadratic 4-node element has been used. But in general there is no analytical solution for higher order elements such as quadratic 8 or 9 node elements. For such elements, in general, it is necessary to use iterative methods such as the Newton-Raphson method to find a solution for inverse mapping. However such numerical computation can be avoided when the higher order elements are parallelograms because, as shown above, the mapping is identical to that of bilinear elements when the mid nodes are centered.

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34 The analytic solution for the inverse mapping for 4 node quadratic elements can be obtained as follows. The mapping equations are: x(r, s) = (4-9) 41),(iiixsrN = 0.25(1+r)(1+s)x 1 + 0.25(1-r)(1+s)x 2 + 0.25(1-r)(1-s)x 3 + 0.25(1+r)(1-s)x 4 y(r, s) = (4-10) 41),(iiiysrN = 0.25(1+r)(1+s)y 1 + 0.25(1-r)(1+s)y 2 + 0.25(1-r)(1-s)y 3 + 0.25(1+r)(1-s)y 4 Expanding and rearranging the above equations are be expressed as follows. x(r, s) = = a 41),(iiixsrN 1 + b 1 r + c 1 s + d 1 rs (4-11) where ) x x x0.25(x d) x x x0.25(x c) x x x0.25(x b) x x x0.25(x a43211432114321143211 y(r, s) = = a 41),(iiiysrN 2 + b 2 r + c 2 s + d 2 rs (4-12) where )y y y 0.25(y d)y y y 0.25(y c)y y y 0.25(y b)y y y 0.25(y a43212432124321243212 If the element is a parallelogram in real space, its two diagonals bisect each other. Therefore, d 1 = 0 and d 2 = 0. Then the following simple analytical solution as shown in Table 4-2 can be found. Table 4-2. Analytical solutions for the 4 node quadratic element mapping (parallelogram) b 1 0 and c 2 0 b 1 = 0, b 2 0 and c 1 0 111bscaxr 211122)(cbscaxbays 111crbaxs 211122)(bcrbaxcayr

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35 Even if the element is not a parallelogram, an analytical solution for the mapping from global coordinates to parametric coordinates can be found even though the solution to the mapping equations is not necessarily unique. Therefore, in this case, the validity of (r, s) should be performed by checking if the obtained r and s satisfy Eq. 4-6. As shown in Table 4-3, the mapping for a non-parallelogram element can have multiple solutions including imaginary solutions. Therefore the 2D element shape in this system is restricted to parallelogram to avoid such ambiguity in mapping. Table 4-3. Analytical solutions for the 4 node quadratic element mapping (non-parallelogram) A 0 A = 0 and B 0 ABsCr 7 BArCs B 0 B = 0 A 0 A = 0 lnlmms242 where m 2 4nl 0 mns lnlmmr242 where m 2 4nl 0 mnr Where Where CbAaAxnCdAcBbmBdl111111 where 11221211221122addaxddyCcddcBbddbA where CcBaBxnCdAbAcmAdl111111 11221211221122addaxddyCcddcBbddbA

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CHAPTER 5 CONSTANT CROSS-SECTION SWEEP It is possible to create commonly used primitive solids such as spheres, cylinders and cones using 3D implicit solid elements. However, the current solid modeling practice involves creating design features by sweep operations such as extrude, revolve and sweep along arbitrary curves. While implementing such operations are easy using B-Rep models, it has not been attempted using implicit surfaces models. In B-Rep models, 2D profiles are defined as a loop of edges and swept along a curve to create sweep features. Surfaces generated by sweeping these edges along any curve are easy to define as parametric surfaces. To accomplish the same using implicit solid elements, the 2D profiles that are swept to create 3D features can be defined by Boolean combination of simple 2D primitives defined as described in the previous chapter. To define a 3D sweep feature using this 2D profile, it is necessary to define 3D implicit solid elements for sweep operations that are derived from any 2D implicit solid element that has been used to define the 2D profile. If the profile is defined using Boolean combination of multiple 2D primitives, then a corresponding 3D primitive is created for each 2D primitive. The 3D sweep feature is then obtained by a similar Boolean combination of the 3D primitives. In the following sections, 3D implicit solid elements are defined for representing solids obtained by extruding, revolving and sweeping a given 2D primitive where constant cross-sections along the parametric t direction are obtained. A more general sweep operation is discussed in the next chapter where variable cross-sections are allowed. 36

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37 5.1 Extrude Elements The extrude element is used to represent the solid obtained by extruding (or sweeping along a straight line) a given 2D primitive defined using an implicit solid element. The direction of extrusion is assumed to be normal to the plane of the 2D primitive. By definition, every section of the extruded solid normal to the direction of extrusion should have a cross-section identical to the 2D primitive being extruded. Assuming that the 2D primitive to be extruded is defined by an implicit solid element with n nodes, the density function within the element can be expressed as follows n2D2D2Diii1(r,s)N(r,s) (5-1) The nodal values of density 2Di are interpolated within the element using polynomial basis functions Figure 5-1 shows the 2D element being extruded and its corresponding extrude element that has twice the number of nodes and its depth D in the parametric direction t is the desired depth of extrusion. Each node i on the front face (t=1) of the element and the corresponding node (i+n) on the back face (t=-1) are set to have equal values of density to ensure that both faces represent the same cross-section geometry. 2DiN( r,s) D Extrusion Figure 5-1. 2D primitive element and the corresponding extrude element The density distribution within the extrude element is interpolated as

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38 231(,,)nExtDExtiiirstN (5-2) The basis functions for interpolating the density within the extrude element are derived such that the cross sectional shape at any t is identical. The basis function at the node i for this element can be defined as follows. 3DiN 321(,,)(1)2DDiiNrsttN for ni.....1 (5-3) 321(,,)(1)2DDji N rsttN for inj The above basis functions ensures that the density distribution within the element is independent of t as shown below ),()1(21)1(21),,(222121212213srNNtNtNtsrDDiniDiniExtiDiniExtiDiExtiniDiExt (5-4) where, the condition used is. DiExtinExti2 This ensures that every cross section from the front to the back face of the extrude element is identical to the geometry represented by the 2D primitive element regardless of the type of the 2D element and the geometry represented by it. The mapping from parametric space to real space can be defined as follows for the extrude element. 231(,,)(,,)XXn E xtDiiirstMrst (5-5) The mapping has to be linear in t to ensure that the solid represented by the element is not distorted in the extrusion direction. In general, it is possible to use the basis functions used for interpolation for mapping also to create an isoparametric element. However, if the edges of the element are straight lines, the mid-edge nodes are at the center of the edges and the center node is at the centroid, then regardless of the number of

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39 nodes in the 2D element it is possible to use the following trilinear interpolation functions (,,)i M rst for extrusion. 1234567811111(1),11(1),11(1)88811111(1),11(1),11(1)8881111(1),11(1)88 M rstMrstMrs t M rstMrstMrstMrstMrst (5-6) An examples of a solid created by extrusion are shown in Figure 5-2 where an ellipse represented using a single 9-node element is extruded to create the solid in Figure 5-2B. (A) (B) Figure 5-2 Solid created by extruding ellipse The profile to be extruded can be created using multiple elements as shown in Figure 5-3 where four elements are used. The first primitive was a rectangle from which an ellipse was subtracted at the bottom. A circle was added to this geometry by union at the top and another smaller concentric circle was subtracted. Upon extrusion, a 3D primitive is created for each of the 2D primitive used to define the profile. These primitives are then combined using the same Boolean operations that were used in the 2D profile to create the solid in Figure 5-3B.

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40 (A) (B) Figure 5-3 Extrusion of profile defined using multiple elements 5.2 Revolve Elements The revolve element is an implicit solid element used to represent the solid obtained by revolving (or sweeping along an arc) a given profile defined using a 2D implicit solid element. The revolved solid element is defined by specifying the 2D element, an axis of revolution and an angle of revolution. It is convenient to use a cylindrical coordinate system to define the nodal coordinates of the element as The cylindrical coordinate system is chosen such that its z-axis is along the axis of revolution of the element. The radial coordinates of the centroid of 2D primitive element is as shown in Figure 5-4. {CTiiiiRZ } 0R 0 R t r s R Z Figure 5-4. Cylindrical coordinate system for revolving As in the extrude element, the density distribution at the front and back element faces as well as any cross-section in between should be identical. Therefore, the

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41 interpolation functions used in the extrude element is also used here to interpolate the density within the revolve element as viniDivNtsrRe213Re),,( (5-7) where, are the densities at the nodes of the revolved element and viRe 3 D iN are the basis functions defined in Eq 5-3. The mapping between the parametric space and the real space can be expressed as: 21(,,)(,,)niiirstMrstCC (5-8) In the above equation, (,,) R Z Ci is the coordinates of an arbitrary point in the element and C i are the nodal coordinates both with respect to a cylindrical coordinate system. The mapping basis functions M could be the same as the interpolation basis functions or when the 2D elements edges are straight lines, one could use the trilinear interpolations functions given in Table 3-2. To express the geometry with respect to a Cartesian coordinate system yet another mapping is required as illustrated in Figure 5-5. Let C be the coordinates of a point in the cylindrical coordinate system and X = (X, Y, Z) be the coordinates of this point in the global coordinate system. (,,)RZ () a b X0 X Z X b a R Z C(R, Z) Y Figure 5-5 Mapping from cylindrical to Cartesian coordinates

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42 The global coordinates with respect to the Cartesian coordinates can be expressed as: 0(,,)(,,)cos((,,))(,,)sin((,,))()(,,)rstRrstrstRrstrstZrstXXaab b (5-9) In the above equation, X 0 is the position vector of the origin of the cylindrical coordinate system while a and b are vectors along the radial and axial directions respectively of the cylindrical coordinate system. Figure 5-6. Examples for revolved solids Two examples of solids created by revolved elements are shown in Figure 5-6, where a sphere is constructed by revolving a semi-circle. The second solid is created by revolving a profile defined using three elements. Two circles are subtracted from a square to create the profile which is revolved 210 o about an axis that is parallel to the edges of the square. 5.3 Sweep Elements The sweep element is used to represent a solid obtained by sweeping a profile represented by a 2D implicit solid element along an arbitrary parametric curve. Therefore, the sweep element is defined by specifying the 2D element, a sweep trajectory as well as an orientation vector d that controls the orientation of the swept profile. The sweep trajectories can be any kind of parametric curves including composite curves. If the trajectory is a composite curve, the number of the 3D elements required to define the swept solid is equal to the number of curves used in the composite curve. The basis ()tR ()t

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43 functions, 3 D iNts),, defined in Eq. 5-3 can be used for this element also to ensure that the density distribution within the element is independent of the parameter t, so that the cross-sections will be identical at any value of t. Therefore, the density distribution in a sweep element can be expressed as lddl SwpiniDiSwpNr213( (5-10) d Sweep trajectory y x z u v n m 0R ()tR l Figure 5-7 Relationship between the global and local coordinate systems For the sweep element also it is convenient to use an intermediate (or local) coordinate system as shown in Fig. 4-7. The local coordinate system is defined using three unit vectors and The vector is tangent to the sweep trajectory can be computed as the derivative of the parametric curve equation R(t). l m n l ttRlR where ()dtdttRR nlmn (5-11) If d is invariant along the sweep trajectory, the sweep will be not be twisted. On the other hand, when d is a function of t and varies along the sweep trajectory, a twisted

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44 sweep will be obtained. In order to create a twisted sweep the user must specify d(t). It is important to ensure that d is not parallel to the curve anywhere along the trajectory. The nodal coordinates of the sweep element is represented with respect to the local coordinate system as L. The mapping between the parametric space and the local coordinates u and v can be expressed as (,,){(,)(,)}Trstursvrst 2121(,)(,)(,)(,)nDiiinDiiiursuNrsvrsvNrs (5-12) where, u i v i are nodal coordinates of the 2D element with respect to the local coordinate system. The mapping between parametric and the global coordinates can therefore be expressed as: 211(,,) or(,,)(,)(,)nnDiiiiiirsttuvrsttuNrsvNrsXR()nmXR()nm 2D (5-13) Figure 5-8 shows two examples created using sweep elements. In the first example, a circle is swept along a cubic spline composed of three Hermite curves. Three sweep elements are therefore used, one for each curve of this composite curve. In the second example a more complex profile defined using nine 2D element is swept along an arbitrary composite curve. The resultant solid is obtained by Boolean combination of the solids represented by individual elements.

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45 Figure 5-8 Examples of solids created using sweep elements

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CHAPTER 6 VARIABLE CROSS SECTION SWEEP As mentioned in Chapter 5, a more general sweep operation can have variable cross-sections. Therefore the variable cross-section sweep using blend solid elements are described in this chapter. Blend elements are used to represent solids created by blending between a pair of 2D primitives (or cross-sections) represented by 2D implicit solid elements. The 2D implicit solid elements used for representing the 2D primitives to be blended are referred to as D profile elements in the following. Since a blended solid can be generated from two or more 2D profile elements, the total number of blend elements used for generating a blended solid depends upon the number of 2D profile elements. In other words, if m 2D profile elements are given then (m-1) 3D blend elements are required to create a solid primitive that blends between these 2D profiles. Unlike extrude, revolve and sweep elements, the shape of the cross-section can vary from one profile to another profile along the t direction in the parametric coordinates. Therefore the intermediate shape from one profile to the next has been studied to obtain a clear understanding of the nature of the blended solid obtained. Two different types of blended solid primitives have been defined in this chapter, namely, straight blend and smooth blend. This classification is based on the surface continuity between the solids defined in adjacent elements. Straight blend is C 0 continuous while smooth blend defines a C 1 continuous blend so that the surface of the primitive defined by these elements is tangent continuous at the interface between 46

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47 neighboring elements. Blend elements also can be classified based on the sweep trajectory along which the 2D profiles are swept. If the sweep trajectory is a straight line, it is called extrude blend element simply denoted by blend element. If the sweep trajectory is a general parametric curve, it is called sweep blend element. To define blend elements, all 2D profile elements used for representing 2D profiles should be the same type of element. In other words, all 2D profile elements should have the same number of nodes per element and the same shape function for each node but they may or may not have the same size and nodal density distributions. In the following section, the detailed description of the blend elements is described. 6.1 Straight Blend Element Two 2D profile elements and the distance between them are required to define a straight blend element. Let the shape density functions of front and back 2D profile elements be and respectively. DF2 DB2 niFiDiDFsrN122),( = (6-1) FnDnFDFDNNN2222121 niBiDiDBsrN122),( (6-2) BnDnBDBDNNN2222121 DiN2 is the 2D basis function at the node i and n is the number of nodes per element in the 2D profile element. Now, density distribution within the straight blend element can be defined by the following shape density function. niiDiSTBlendNtsr213),,( (6-3)

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48 The basis functions for interpolating the nodal densities are designed to give linear interpolation in the parametric direction t that is the desired direction of depth. The following equation shows the basis function at the node i. DiN3 ),()1(21),,(23srNttsrNDiDi for ni.....1 at the front face of the element ),()1(21),,(23srNttsrNDiDj for inj at the back face of the element (6-4) The mapping from the parametric space to the real space for the straight blend element can be defined as niDiDitsrMtsr2133),,(),,(XX (6-5) Di3X and are the nodal coordinates and the mapping basis function at the node i respectively. As in the previous chapter, it is assumed that the mid-edge nodes are at the mid point of the edges and that the center node is at the centroid so that the trilinear mapping functions defined in Eq. 5-6 can be used. DiM3 The cross-sectional shape at the common face between elements is identical to the 2D primitive used for creating the straight blend element. Therefore the blended solid defined by the straight blend element has the following properties. Property 6.1-1: The blended solid represented using the straight blend elements satisfies C 0 continuity at the interface between each neighboring element.

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49 Element BElement ANeighboring face[t= -1 at Element A, t= 1 at Element B] t = -1 atElement Bt = 1 atElement Abbb Figure 6-1. Two neighboring straight blend elements having quadrateral 9 node elements as 2D profile Proof: It can be verified by showing that the density distributions at t = -1 at the element A and t = 1 at the element B are identical to the density distribution of the 2D profile element used for the common face. Therefore the shape of interest at the common face between two neighboring elements is same as the 2D profile used. DBAniBiDiSTBlendAsrNsr212),()1,,( at the element A DFBniFiDiSTBlendBsrNsr212),()1,,( at the element B (6-6) DFBDBA22 because the same profile element is used for representing the back face of the element A and the front face of the element B. As mentioned earlier, the 2D profile elements may or may not have the same density distribution. If the density distributions of the front and back 2D profile elements are identical, the surfaces connecting the front and back 2D profiles have the following property.

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50 Property 6.1-2. The surface connecting the front and the back 2D profiles in the straight blend element is composed of straight lines that connect points having the same r and s but different t value (i.e. t = -1 and t =1 respectively) only if. BiFi X(r*, s*, -1)X(r*, s*, 1)X(r*, s*, t) Front face where t=1 Back face where t = -1b b Figure 6-2. A straight blend element having quadratic 9 node elements as 2D profiles Proof: If the density distribution for the front and the back 2D profile elements are the same, the shape density function for the straight blend element can be obtained by applying the condition as follows. BiFii niiDiniiDiSTBlendsrNttsrN1212),()1(21)1(21),( (6-7) Eq. 6-7 shows that the cross-sectional shapes along the parametric direction t are invariant with respect to the t in the parametric space. Suppose (r ,s ) is the parametric coordinates of a point on the boundary so that (boundary density) then X(r bDsr),(**2 s t) is the position vector of a point on the curve along the surface of the solid connecting points on the front and back faces that share the same parametric coordinates (r ,s ). It would now be shown that this curve is a straight line.

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51 nn2D**F2D**Biiii1i1nn**2D**F2D**Biiii1i1nn2D**F2D**Biiii1i111N(r,s)(1t)xN(r,s)(1t)x22x11X(r,s,t)yN(r,s)(1t)yN(r,s)(1t)y22z11N(r,s)(1t)zN(r,s)(1t)z22 iii BFttXX)1(21)1(21 (6-8) In the above equation, and are the nodal coordinates at the front and the back face of the straight blend element respectively and and are defined as ),,(FiFiFizyx ),,(BiBiBizyx FX BX niFiDiniFiDiniFiDiFzsrNysrNxsrN1**21**21**2),(),(),(X and (6-9) niBiDiniBiDiniBiDiBzsrNysrNxsrN1**21**21**2),(),(),(X Eq. 6-8 represents a line equation connecting the point and From Eq. 6-7 the density values at and are same and the densities through the line is also same. Therefore it is verified that the surface is composed of straight lines connecting two points having the same r and s and different t = and t = 1 respectively. Such a surface is known as a lofted surface (or ruled surface) [Filip and Ball, 1989]. Intuitively, it is clear that in general when the 2D profile elements defining the blend element have different density distributions the blending surfaces are not guaranteed to be lofted surfaces. FX BX FX BX Investigating cross section shapes of the blend element along the parametric t direction, the following property can be obtained.

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52 Property 6.1-3. The cross-sectional shape of the straight blend element at a certain parameter t is a parallelogram if the mid-edge nodes are at the center of each edge and mid-face node is at the centroid of the face. t =1 t = -1t = t* r = 1s = 1r = -1s = -1r = 1s = 1r = -1s = -1 X(1,1, t*)X(-1,1, t*)X(-1,-1, t*)X(1,-1, t*)12348567 Figure 6-3: Cross section at a certain t in the straight blend element Proof: To prove that the cross-section is a parallelogram it is first shown that the edges of the cross-section are straight lines. It can be assumed that the mapping basis functions are tri-linear regardless of the number of nodes in the element if the mid-edge nodes are at midpoint of the edges and mid face nodes are at the centroid of the face. Therefore the mapping equation can be expressed as 8ii1(r,s,t)M(r,s,t)X iX (6-10) In the above equations Mi are the trilinear basis functions and X i are the nodal coordinate points in the global coordinate system. Expanding the above equation by substituting the trilinear basis functions for M i (r,s,t), yields,

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53 87654321)1)(1)(1(81)1)(1)(1(81)1)(1)(1(81)1)(1)(1(81)1)(1)(1(81)1)(1)(1(81)1)(1)(1(81)1)(1)(1(81XXXXXXXXXtsrtsrtsrtsrtsrtsrtsrtsr (6-11) At any cross-section of the element t is constant so it will be denote as t The equation of the boundaries of the element at this cross-section can be obtained by setting parametric coordinates r and s to or 1. Consider the edge where r = 1, the equation of the edge, X(1,s,t ), can be expressed as 8541*)1)(1(41*)1)(1(41*)1)(1(41*)1)(1(41*),,1(XXXXXtststststs (6-12) Rearranging, 8451*)1(21*)1(21)1(21*)1(21*)1(21)1(21*),,1(XXXXXttsttsts Since t X 1 ,X 5 ,X 4 and X 8 are constants, it is clear that the equation of X is that of a straight line where only s is a variable. Similarly it can be verified that other three edges defined by r = 1, s = -1 and s = 1 are also straight lines. Now to show that the quadrilateral defined by these four straight lines is a parallelogram it needs to be verified that the diagonal of this quadrilateral bisect each other. Let four vertices of the cross-section of the element at a given t be ,,and respectively. *1X *2X *3X *4X 5*1**1*)1(21)1(21),1,1(XXXXttt 6*2**2*)1(21)1(21),1,1(XXXXttt

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54 7*3**3*)1(21)1(21),1,1(XXXXttt 8*4**4*)1(21)1(21),1,1(XXXXttt (6-13) The mid point of each diagonal can be represented by ))(1(21))(1(2121275*31**3*1XXXXXXtt ))(1(21))(1(2121286*42**4*2XXXXXXtt (6-14) From the fact that 2D profile elements are parallelograms, the following equations can be obtained. 224231XXXX and 228675XXXX (6-15) Applying the above conditions to Eq.6-14, the following relationship can be obtained. 22*4*2*3*1XXXX (6-16) Therefore the midpoints of the two diagonals are the same thus proving that the cross-section is a parallelogram. This is a very important property because it can be used for the mapping from the global coordinates to the parametric coordinates. Once it is verified that the cross-section shape of the straight blended element at a certain parametric coordinate t* is a parallelogram, then two-dimensional mapping techniques can be used for finding the parametric coordinates r and s. Examples of solids created by blending 2D profiles are shown in the following figure where four 2D profiles are used for both examples. Figure 6-4A shows the

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55 blending between circular shapes and figure 6-4B shows the blending between circular and rectangular shapes. (A) (B) Figure 6-4. Examples of straight blend using blend elements 6.2 Smooth Blend Element Smooth blend elements construct a C1 continuous blended solid between 2D profile elements defined along a straight line. The shape density function for the smooth blend element can be defined by nitiDtiiDiSMBlendNNtsr2133)(),,( (6-17) i and ti are the nodal density and its corresponding tangent component at the node i respectively. The basis functions for interpolating the nodal densities are designed to give Hermite interpolation in the parametric direction t. The following shows the basis function and at the node i. D iN3 DtiN3 DiN3 = 322212213ttNDi = for i 12HNDi n.....1 DitN3 = 3222121ttNDi = for i 22HNDi n.....1 (6-18)

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56 DjN3 = 3222122131ttNDi = for 32HNDi in j DtjN3 = 3222121221tttNDi = for 42HNDi in j A composite Hermite curve is guaranteed to be C 1 continuous regardless of the tangent at the nodes. However, to set the tangent components at each node automatically, the following two additional conditions are used. First, the second derivative at the end each element should be equal to that at the beginning of the adjacent element. Second, the second-order derivatives are set to zero at both ends of the primitive. According to these conditions, the following linear equations are obtained, by which an automatic method for setting the tangent components can be achieved. The following equation is a matrix form for N simultaneous linear equations. Therefore, the tangent components can be computed by solving the following equation. 210001410:0141:0......::........::......::.......0:1410:01410........0012 ''1'3'2'1:::::NNRRRRR = (6-19) 122413123333:::::333333NNNNRRRRRRRRRR In the above equation, iR i and iR ti In the smooth blend element, the mapping from the parametric space to the real space can be performed as )),,(),,((),,(332133DtiDtiniDiDitsrMtsrMtsrXXX (6-20)

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57 Di3X and are nodal coordinates and its tangent components at the node i. and are mapping basis functions that are identical to Dti3X DiM3 DtiM3 DiN3 and DtiN33X respectively for this element. is computed from Eq. 6-19 setting Dti3X D i iR and 3X D iR ti The following property is proved to show that the solid created using the smooth blend elements has C 1 continuity along the parametric coordinate t. Property 6.2-1. The blended solid using the smooth blend elements satisfies C 1 continuity at the interface between adjacent elements. Element BElement ACommon face[t= -1 at Element A, t= 1 at Element B] t = -1 atElement Bt = 1 atElement Ab )1,,()1,,(srsrEBEA Figure 6-5. Two neighboring smooth blend elements having quadratic 9 node elements as 2D profile Proof: This property can be verified by showing that the surface normal vector is continuous at the interface between each neighboring element. Suppose the face of interest is the back face (i.e. t = -1) in the element A denoted by EA and the front face (i.e. t = 1) in the element B denoted by EB as shown in figure 6-5. The surface normal vector is computed by gradient of the shape density function or normal to the element surface in this system. The following is the formula for computing the gradient of the shape density function with respect to the global coordinates x, y, z.

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58 tsrzyx1J Where, the Jacobian matrix tztytxszsysxrzryrxJ (6-21) Let the shape density function at the element A and the element B be EA and EB respectively, then BEAtiBEAiFEAtiFEAiniDiEAtHtHtHtHsrNtsr)()()()(),(),,(432112 BEBtiBEBiFEBtiFEBiniDiEBtHtHtHtHsrNtsr)()()()(),(),,(432112 (6-22) Since both element A and element B are sharing the 2D profile element at the common face, the density distributions at the back face of the element A and the front face of the element B are identical. Therefore, FEBiBEAii and (6-23) FEBtiBEAtiti Differentiating EA and EB with respect to r, s and t and thereafter substituting for the parameter t at the element A and 1 for the parameter t at the element B into Eq. 6-22 and applying Eq. 6-21. rsrrsrNrsrEBniiDiEA)1,,(),()1,,(12 ssrssrNssrEBniiDiEA)1,,(),()1,,(12 tsrsrNtsrEBnitiDiEA)1,,(),(21)1,,(12 (6-24) Similarly, the above procedure for x, y and z instead of can be applied. Now, let x, y and z be and with respect to the corresponding elements. Then EAx EAy EAz EBx EBy EBz

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59 BEAtiBEAiFEAtiFEAiniDiEAxtHxtHxtHxtHsrNtsrx)()()()(),(),,(432112 BEBtiBEBiFEBtiFEBiniDiEBxtHxtHxtHxtHsrNtsrx)()()()(),(),,(432112 (6-25) Where and FEBiBEAiixxx FEBtiBEAtitixxx Differentiating and with respect to r, s and t and then substituting for the parameter t at the element A and 1 for the parameter t at the element B into Eq. 6-25. EAx EBx rsrxxrsrNrsrxEBniiDiEA)1,,(),()1,,(12 ssrxxssrNssrxEBniiDiEA)1,,(),()1,,(12 tsrxxsrNtsrxEBnitiDiEA)1,,(),(21)1,,(12 (6-26) Following the same procedure for y and z, the similar relation to Eq. 6-26 for y and z can be found. From Eq.s 6-24 and 6-26, yields, )1,,()1,,(21srsrEE (6-27) Therefore Eq. 6-27 proves the property 6.2-1. If the density distributions between the front and the back 2D profile elements are identical, the surfaces connecting the front and back 2D profiles have the following property. Property 6.2-2. The surfaces connecting the front and the back 2D profiles in the smooth blend element are composed of Hermite curves that connect points having the same r and s but different t value (i.e. t = -1 and t =1 respectively) only if. BiFi

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60 X(r*, s*, -1)X(r*, s*, 1)X(r*, s*, t) Front face where t=1 Back face where t = -1bb Figure 6-6. A smooth blend element having quadratic 9 node elements as 2D profile Proof: If the density distributions for the front and the back 2D profile elements are same (i.e. ), it implies that all tangential components BiFi ti are zeros according to Eq.6-22. Therefore the density function can be simplified as follows. niBiDiniFiDiSMBlendttsrNttsrN132213222122131),(212213),(BiFii Applying the condition then niiDiSMBlensrN12),( (6-28) The above equation shows that the cross-sections along the direction of t in the parametric coordinate system are invariant. Let (r s ) be the parametric coordinate of a point on the boundary of the solid so that (boundary density), then density value at every point on a straight line connecting (r bDsr),(**2 s -1) and (r s 1) in the parametric coordinate system is b because Eq. 6-28 shows that density is invariant with respect to t. Therefore, this line is guaranteed to be on the surface of the solid. The parametric equation of the line mentioned above in the global coordinate system is

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61 BtiBiFtiFiniDiBtiBiFtiFiniDiBtiBiFtiFiniDiztHztHztHztHsrNytHytHytHytHsrNxtHxtHxtHxtHsrNzyxtsr)()()()(),()()()()(),()()()()(),(),,(43211**243211**243211**2**X (6-29) BtBFtFtHtHtHtHXXXX)()()()(4321 This is a Hermite curve equation where niFiDiniFiDiniFiDiFFFFzsrNysrNxsrNzyx1**21**21**2),(),(),(X and (6-30) niFtiDiniFtiDiniFtiDiBtBtBtFtzsrNysrNxsrNzyx1**21**21**2),(),(),(X niBiDiniBiDiniBiDiBBBBzsrNysrNxsrNzyx1**21**21**2),(),(),(X and (6-31) niBtiDiniBtiDiniBtiDiBtBtBtBtzsrNysrNxsrNzyx1**21**21**2),(),(),(X This curve is also a cardinal spline if it satisfies the following equation. FEABEBBEBttFEAtXXXXX34 (6-32) Where FEBtBEAttXXX This equation can be written as follows. niFEAiBEBiDiniFEAiBEBiDiniFEAiBEBiDiniBEBtitiFEAtiDiniBEBtitiFEAtiDiniBEBtitiFEAtiDizzsrNyysrNxxsrNzzzsrNyyysrNxxxsrN1**21**21**21**21**21**2),(),(),(34),(4),(4),( Or niFEAiBEBiDiniBEBtitiFEAtiDisrNsrN1**21**2),(34),(XXXXX (6-33)

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62 Since the solid element is defined such that FEAiBEBiBEBtitiFEAtiXXXXX34 (see Eq. 6-19), the above equation is automatically correct. Therefore when density distribution is identical for front and back faces the smooth blend primitives have C 2 continuity along the t direction at the common face between neighboring solid elements because curves on its surface in the parametric direction t are Cardinal splines. However, in general when 2D profile elements defining the smooth blend element have different density distributions, the resulting blended solids boundary is not composed of cardinal splines. Investigating cross-sectional shapes of the smooth blend element along the parametric direction t, the following property can be obtained. Property 6.2-3. The cross-section shape of the smooth blend element at a certain parameter t is a parallelogram if the mid-edge nodes are at the center of each edge and mid-face node is at the centroid of the face. E2E1Common face[t= -1 at element A t= 1 at element B] t = -1 at element Bt = 1 atelement A t = t* r = 1s = 1r = -1s = -1 X(1,1, t*)X(-1,1, t*)X(-1,-1, t*)X(1,-1, t*) 1143258761011129 Figure 6-7. Cross section at a certain t* in smooth blend elements

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63 Proof: To prove this property, it is first verified that it is true in the smooth blend element. Initially it is assumed that 4 node quadrilateral elements are used for the 2D profile elements. The mapping equation as shown in figure 6-7 is 81)),,(),,((),,(ititiiitsrNtsrNtsrXXX (6-34) Consider the edge where r = 1, for any cross-section t=t The points on this edge, X, can be written as 8454421283534111)()1(21)()1(21)()1(21)()1(21)()1(21)()1(21)()1(21)()1(21),,1(tttttHstHstHstHstHstHstHstHstsXXXXXXXXX (6-35) From Eq. 6-19, it is known that, 1551332XXXXtt 4884332XXXXtt 19951334XXXXXttt 4121284334XXXXX ttt According to the above relationships, X(1,s,t) can be simplified as 8483424154531211)()()()()1(21)()()()()1(21),,1(tttttHtHtHtHstHtHtHtHstsXXXXXXXXX (6-36) Since t, X 1 X 5 X 4 X 8, X t1 X t5 X t4 and X t8 are constants, therefore the final form of X(1,s,t) is a line equation where the variable is s. Similarly it can be verified that other three edges defined by r = 1, s = -1 and s = 1 are also straight lines. This proves that the

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64 cross sectional shape of the blend element at a certain t* is a quadrilateral. This quadrilateral is a parallelogram if its diagonals bisect each other as shown below. Let four vertices of the cross-sectional shape be ,,and respectively. *1X *2X *3X *4X 54531211*1*)()()()(),1,1(tttHtHtHtHtXXXXXX 64632221*2*)()()()(),1,1(tttHtHtHtHtXXXXXX 74733231*3*)()()()(),1,1(tttHtHtHtHtXXXXXX 84834241*4*)()()()(),1,1(tttHtHtHtHtXXXXXX (6-37) The midpoints of the two diagonals are ))(())(())(())((212754753312311*3*1tttttHtHtHtHXXXXXXXXXX ))(())(())(())((212864863422421*4*2tttttHtHtHtHXXXXXXXXXX (6-38) Again from Eq. 6-19, the followings can be obtained. 1551332XXXXtt 2662332XXXXtt 3773332XXXXtt 4884332XXXXtt The above equations can be combined to yield the following equations. )(3)(3)()(231757531XXXXXXXX tttt )(3)(3)()(242868642XXXXXXXX tttt (6-39) From the fact that the 2D profile elements used to define the smooth blend are parallelograms, 224231XXXX and 228675XXXX (6-40) Therefore, the following equation can be obtained from Eq. 6-39.

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65 )()(2)()(286427531ttttttttXXXXXXXX (6-41) The following is also satisfied. 19951334XXXXXttt 2101062334XXXXX ttt 3111173334XXXXXttt 4121284334XXXXX ttt The above equations can be combined and yield the following equations. )(3)(3)()(4)(311191197531XXXXXXXXXX tttttt )(3)(3)()(4)(42121012108642XXXXXXXXXX tttttt (6-42) From the fact that 2D profile elements are parallelograms, 224231XXXX and 221210119XXXX (6-43) Therefore, the following equation from Eq. 6-42 and 6-43 can be obtained. )()(4)()()(4)(121086421197531ttttttttttttXXXXXXXXXXXX (6-44) Applying Eq. 6-41 and Eq. 6-44 to the Eq. 6-38, yields 22*4*2*3*1XXXX (6-45) Therefore the cross-section is a parallelogram since Eq. 6-47 shows that the midpoints of the two diagonals are coincident. So far, it is assumed that 4 node elements are used as 2D profile elements to verify property 6.2-3. This is also true when 9 node elements or others are used if their mapping equations are equal to the mapping equation at 4 node elements as explained below. Let the mapping equation of the smooth blend element having 4 node profile elements be X 4

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66 41)4(44241314)),(),(()),(),((),,(iitiiiitiiiisrNHsrNHsrNHsrNHtsrXXXXX (6-46) The following shows the mapping function at the smooth blend element where quadratic elements having n nodes are used for 2D profile elements. ninitiniinitiiiinsrNHsrNHsrNHsrNHtsr1)(42131)),(),(()),(),((),,(XXXXX (6-47) If the mapping equation in the 2D profile element having n nodes is same as the mapping equation in the 4 node quadrilateral element, i.e., if (as proved in Chapter 4 for 9 node elements, Eq. 4-8), the following is also true. 411(,)(,)niiiiiNrsNrsX iX ),,(),,(4tsrtsrnXX (6-48) This property can be used for the mapping from the global coordinates to the parametric coordinates. Once it is verified that the cross-section shape of the smooth blend element at a certain parametric coordinate t* is a parallelogram, then two-dimensional mapping techniques can be used for finding the parametric coordinates r and s. Examples of solids created by blending 2D profiles are shown in the following figure where four 2D profiles are used for both examples. Figure 6-8A shows the blending between circular shapes and figure 6-8B shows the blending between circular and rectangular shapes.

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67 (A) (B) Figure 6-8. Examples of Smooth blend using blend elements 6.3 Sweep Blend The sweep blend element is used to represent a solid obtained by sweeping 2D profiles represented by 2D implicit solid elements along an arbitrary parametric curve. Therefore, the sweep blend element is defined by specifying the 2D profile elements, a sweep trajectory as well as an orientation vector that controls the orientation of the swept profile. The detailed description regarding sweep trajectories is explained in Chapter 5. ()tR )(td Two different types of blending functions, straight and smooth blending can be applied to sweep blend too. Therefore all properties illustrated already in the previous sections are also applicable to sweep blend. Since the nodal coordinates of the sweep blend elements are represented with respect to the local coordinate system as L(r,s,t) ={u(r,s,t) v(r,s,t) t} T the mapping between the parametric coordinates and the local coordinates u and v can be expressed as niiitsrNutsru1),,(),, (

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68 niiitsrNvtsrv1),,(),,( (6-49) where, u i v i are nodal coordinates of the sweep solid element with respect to the local coordinate system. N i (r,s,t) are basis functions for both the mapping from the parametric coordinates r, s and t and interpolating nodal densities. Therefore the basis functions are different with respect to the blend types such as straight blend and smooth blend. Examples of solids created using sweep blend element are shown in the following figure where four 2D profiles are used for both examples. Figure 6-9A shows the sweep straight blending between rectangular shapes and figure 6-9B shows the sweep smooth blending between circular shapes. (A) (B) Figure 6-9. Examples of Sweep Blending A) Sweep straight blend B) Sweep smooth blend

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CHAPTER 7 CSG REPRESENTATION USING IMPLICIT ELEMENTS CSG representation scheme is used for representing a solid in this solid modeling system where implicit solid elements are used for representing primitives instead of half spaces. The final solid is constructed by combining simple primitives using regularized Boolean operations to define a solid as a closed region in space. It is necessary to compute points and surface normal vectors on the boundary of the solid for graphically displaying it. The following sections, describe the CSG tree structure used in this system, how to compute surface normal vectors, and how to perform set-membership classification. 7.1 Surface Normal Vectors To graphically display primitives defined using implicit solid elements, it is necessary to evaluate points on the surface and to compute surface normal vectors at these points. The method for evaluating points on the surface is explained in chapter 9. Here, the method for computing the surface normal vector at a given point on a primitive represented by an implicit solid element is described. To compute the surface normal vector at a given point on the boundary of a solid represented using an implicit solid element, the point has to be first classified as inside the element () or on the boundary of the element (r). If the point is inside the implicit solid element, the gradient of the density function is computed since it is a vector normal to the surface at the point of b,1r1,1s1,1t11 or t=1 1 or s= 69

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70 interest. The gradient of with respect to the global coordinates (x, y, z) can be computed as: tsrzyx1J (7-1) Where tztytxszsysxrzryrxJ (7-2) The [J] matrix or the Jacobian matrix can be easily obtained since the mapping expresses x, y and z as functions of r, s and t. For instance, the first component of the Jacobian matrix can be expressed as where ntiitsrNxx1),,(. rtsrNxrtsrNxrtsrNxrtsrNxrxnnniii),,(),,(),,(),,(22111 (7-3) Similarly, the other components of the first column are stsrNxstsrNxstsrNxstsrNxsxnnniii),,(),,(),,(),,(22111 ttsrNxttsrNxttsrNxttsrNxtxnnniii),,(),,(),,(),,(22111 (7-4) The second and third columns of the Jacobian matrix can be computed by replacing x with y and z in Eq. (7-3) to (7-4) respectively.

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71 Note that the gradient points towards the direction of increasing density that is towards the inside of the solid. Therefore, the surface normal is set in the opposite direction as n The actual equations for finding each component of the Jacobian matrix depend upon the mapping functions of r, s and t. Therefore, the solid elements having different mapping functions need to be treated accordingly. Since extrude, revolve and sweep elements have their own mapping functions that are different from each other, they have their own corresponding equations for finding the components of the Jacobian matrix as follows. In extrude elements including extrude blend elements, Eq. 7-3 and Eq. 7-4 can be used directly because the global coordinates (x, y, z) are just functions of the parametric coordinates (r, s, t). Therefore each component of the Jacobian matrix can be computed by just differentiating the mapping function with respect to the parametric coordinates (r, s, t). However revolve elements use the cylindrical coordinate system whereas the extrude elements use the Cartesian coordinates system. Therefore one more step is required to find the surface normal vectors for revolve elements. The following is an example where the first components of the Jacobian matrix is computed. rZZxrxrRRxrx (7-5) Here Rx x and Zx can be derived from Eq. 5-9 explicitly. sincosxxcaRx cossinRcRaxxx and xbZx (7-6) Since R, and Z are represented by the interpolation using the nodal basis functions, their differentiations are

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72 rtsrNRrRniii),,(1 rtsrNrniii),,(1 (7-7) rtsrNZrZniii),,(1 In sweep elements, the following equations are used for computing the each component of the Jacobian matrix where the notations used in Eq. 7-8 are followed from chapter 5. niiDiniiDivdrsrdNudrsrdNdrdzdrdydrdxdrtsrd1212),(),(),,(mnP niiDiniiDivdssrdNudssrdNdsdzdsdydsdxdstsrd1212),(),(),,(mnP (7-8) dtddtdzdtdydtdxdttsrdR(t)P),,( If the point of interest is on the boundary of the element, the surface normal vector is equal to the vector normal to the element face on which the point is located. The position vector of an arbitrary point within the element is expressed as follows ),,(tsrR

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73 niii1niii1niii1xM(r,s,t)x(r,s,t)yyM(r,s,t)zzM(r,s,t)R (7-9) The six faces of a hexahedral element are represented as , , ),,1(tsR ),,1(tsR ),1,(trR ),1,(trR )1,,(srR )1,,( srR,,1(tsR respectively. Since these are parametric equations of surfaces, the normal vector for any point on these faces can be easy computed as the cross product of the tangent vectors along the parametric directions. For example, normal vector to the face is ) (1,s,t)(1,s,t)stRRn (7-10) Similarly, the normal vectors can be computed on the other five faces. Table 7-1 shows the normal vectors on all six faces of the element. Table 7-1 Surface normal vectors on the boundary of the element Element Face Normal Vector Element Face Normal Vector ),,1(tsR ttssts),,1(),,1(RR ),,1(ts R ststts ),,1(),,1(RR ),1,(trR rtrttr),1,(),1,(RR ),1,(tr R ttrrtr ),1,(),1,(RR )1,,(srR ssrrsr)1,,()1,,(RR )1,,( srR rsrssr )1,,()1,,(RR Surface normal is ambiguous for points along the edges and vertices of the implicit solid element unless a face of the element is also specified whose normal is to be computed at the point.

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74 7.2 Set membership classification within an element An important requirement for any unambiguous solid model representation is that one should be able to classify a given point as being inside, on, or outside the solid. Algorithms for such point set member classification are non-trivial for B-Reps. For traditional implicit surfaces, one can use the value of the implicit surface function to determine which side of the surface a point lies on. However, for primitives defined using implicit solid elements, the density function is expressed as a function of the parametric coordinates and therefore it is necessary to first map a point from the real space into the parametric space to evaluate the density. The mapping between those two spaces is typically non-linear. If the given point is p X then the following equations for the parameters p r p s and p t that corresponds to this point need to be solved. p iipppM(r,s,t)XX (7-11) If the mapping basis functions, are nonlinear three simultaneous non-linear equations need to be solved. Since it is not always possible to obtain analytical solution for such equations it is necessary to use numerical methods such as Newton-Raphson method to find a solution. Clearly, these equations will be easier to solve if simple basis functions for the mapping are used. For example, when a 9-node element is used, the interpolation basis function is bicubic, but one could use a simpler bilinear (4-node) basis functions for the mapping if the element edges are not distorted into curves and mid-nodes are mid points of each edge. Moreover the mapping function can be solved analytically to find the parametric coordinates from the given global coordinates as shown in Chapter 4. ipppM(r,s,t)

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75 An analogous set of mapping equations for (x, y, z) can be derived for the 8-node hexahedral element where trilinear basis functions are used. Even though for the general case these equations are nonlinear, they reduce to linear simultaneous equations for elements whose opposing faces are parallel. Therefore, if the shape of the element in the real space is a parallelogram in 2D or a parallelepiped in 3D then the mapping is linear and easy to perform in either direction. However, in general, three nonlinear simultaneous equations need to be solved numerically to compute the parametric coordinate. To avoid such numerical computation 3D problems are reduced to 2D problems that can be solved analytically. This is based on the idea that the cross-sections of the implicit solid element along the sweep direction are parallelograms and the mapping from the real space to the parametric coordinates of this parallelogram is bilinear. All predefined 2D primitives are parallelograms so that the mapping equations can be solved analytically. Therefore the first step in mapping is to find the parametric value t for the given point. When the sweep trajectory is a straight line, the parameter t can be easily computed by comparing the distance from the given point and the element front face and the depth of the element. Similarly, when the sweep trajectory is an arc, the parameter t can be computed by comparing the angle between the given point and the element front face and the total angle of revolution of the element. In the case of sweep elements including sweep blend elements where the sweep trajectories are arbitrary parametric curve equations, the nonlinearity in the mapping is introduced by the sweep trajectory. The parameter t can be computed as the optimal solutions obtained by minimizing ()tpRX that is by finding the closest point on the

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76 trajectory R(t) from the given point X p where the R(t ) should satisfy the following equation. 0))(()(**PtttXRR (7-12) Now, from the parameter t obtained from the above procedure, a parallelogram having the four vertices such as X(1, 1, t ), X(-1, 1, t ), X(-1, -1, t ) and X(1, -1, t ) can be built. Thereafter analytical method can be used to find the parametric coordinate r and s. By using this technique, set-membership classification is fast and efficient in this system. After finding the parametric coordinates (r, s, t) from the given global coordinates (x, y, z), the given point can be easily classified as inside, outside and on the solid with respect to primitive by computing the shape density value at the point. Figure 7-1 shows the procedure for set-membership classification with respect to an element as a flow chart. Yes Yes Yes N o N o N o Outside N o Yes End Inside On the boundary r,s or t = -1 or1 = b Compute = (r,s,t) b -1 r,s,t 1 Mapping from (x,y,z) to (r,s,t) Start from a given point (x,y,z) Figure 7-1 Flow chart for set-membership classification in an individual solid element

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77 7.3 Constructive Solid Geometry using Implicit Solid Elements As mentioned earlier, CSG representation is a well-established technique for representing solid models and is used to construct solid models by Boolean combination of primitives that are represented either as B-Rep or using implicit half spaces. The same approach can be used with primitives represented by the implicit solid elements. An advantage of using the implicit surfaces is that set membership classification is computationally less expensive than with B-Rep. Furthermore, there is no need for expensive boundary evaluation algorithm, which is needed in the hybrid approach to automatically construct the B-Rep model of the solid represented by the CSG tree. This step is necessary in the hybrid approach because the B-Rep model of the Boolean result is needed for creating the graphical display and also for subsequent operations such as set membership classification. However, if the primitives used in the CSG tree are represented using implicit solid elements, it is possible to directly compute the solids properties, generate graphical display and perform set membership classification based on the procedural description of the solid embodied in a CSG tree. The following figure shows the CSG tree data structure used in this system. (a) 2D CSG tree (b) 3D CSG tree Primitive Node Boolean Node Figure 7-2 CSG tree data structure

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78 The CSG tree shown in Figure 7-2 illustrates how a solid is represented in this system. Every leaf node represents a primitive defined by implicit solid elements and the branch node represents a Regularized Boolean type such as union, intersection and subtraction (respectively). Before performing Boolean operations the two participating solids need to be positioned correctly with respect to each other. This could be achieved by translations and rotations specified by the user but for dimension driven editing the required transformation must be computed based on specified positioning constraints. In this implementation, a sequential constraint imposition method [Kumar, 2000] was used to compute the transformation matrix required to impose position constraints between solids. Typically, position constraints are specified between vertices, edges and faces of a solid if the solid is represented using a B-Rep model. Even though such entities are not available in implicit solid models, the nodes, edges and faces of the implicit element itself serve as useful datum that can be used for specifying constraints. // Figure 7-3 is a simple example where the 2D profile is obtained by the union of a rectangle and a circle followed by the subtraction of an ellipse. Figure 7-4 shows a 3D example. U Figure 7-3 2D Boolean operation in CSG

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79 Figure 7-4 3D Constructive Solid Geometry Tree For a primitive solid defined using a single element, the shape density value at a point can be computed as used to classify the point as inside the solid, on the boundary of the solid or outside the solid. When many primitives have been combined using Boolean operations, a procedure for classifying points with respect to the Boolean results are required. In order to apply the same procedure for the Boolean result the shape density at every point within the solid including regions where multiple primitives may intersect needs to be defined. This can be accomplished by using the following definition that is commonly used in the literature on level set methods. ),max(BABA Union ),min(BABA Intersection (7-13)

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80 ),min(BABA Subtraction In the above definitions, A B is the density at any given point within the solid defined as the union between two solids A and B whose shape density values are A and B respectively. It can be easily verified that A B will be greater than the level set value at every point within the solid defined by the union. It is equal to the level set value at the boundary and less than equal to outside the solid. Similar definitions are given above for the intersection and difference between solids. Another way to define density at every point within a solid is by using R-functions [Shapiro, 1998], [Shapiro, Tsukanov, 1999] which can be used to define the density for the solid defined using Boolean operations. The sign of such a function depends only on the sign of its arguments and not their magnitude. This property can be used to define density values at points where two or more primitives intersect. However, to use R-functions it is necessary that every point on the boundary of the solid have value equal to the level set value b In implicit solid elements, the solids boundary can be the same as element boundary where the density need not be the level set value. Therefore, to use R-function method the definition of the implicit solid element will have to be modified accordingly. In this implementation, R-functions to define density for Boolean solids were not used. As mentioned Chapter 2, when the given point is on both the base and the dependent solids, it is difficult to determine its set membership with respect to the regularized Boolean result by simply comparing its set membership with respect to base and dependent solids. In order to apply the rules of regularized Boolean operations, it is necessary to study the neighborhood of the given point to check whether it is on the

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81 boundary or the interior of the solid. Alternately, one can compare the direction of surface normal of the solids between which the regularized set operation is being applied [Mantyla, 1988]. This approach can be applied recursively to perform set membership classification for arbitrarily large CSG trees. By applying this procedure the density functions only at the primitive level are used. As mentioned in Chapter 2, the neighborhood model enables us to classify points with respect to the regularized Boolean when multiple normal vectors are involved. From a theoretical point of view, the neighborhood model at a given point is easy to understand but it is very difficult to construct an algorithm based on this model since there are infinite points in a neighborhood that needs to be checked. In this system, a graphical method for constructing the neighborhood model is proposed and implemented. This method is based on the assumption that the maximum number of normal vectors involved at a given point is three in the level of solid elements and it is convex not concave. Therefore triangles from the normal vectors that represent half spaces can be constructed. The detailed description for constructing triangles from the surface normal vectors is explained in the following. In a solid element, triangles can be constructed from the surface normal vectors as shown where a face of the solid has one normal vector, an edge of the solid has two normal vectors and a vertex of the solid has three normal vectors respectively.

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82 nC2nC1nB3nB2nB1nA1nA1nB1nB3nB2nC1nC2 Figure 7-5. surface normal vectors and corresponding triangles When there is only one surface normal vector at a given point, a triangle normal to this vector can be constructed by the following procedure. n ABCabcP Figure 7-6: Constructing a triangle from one normal vector In the Figure 7-6, n is the unit surface normal vector ( 1n ). a, b and c are unit vectors perpendicular to the surface normal vector n. a can be chosen from any arbitrary vector perpendicular to the surface normal n. Therefore, a should satisfy the following condition. 0na and 1a Then point A can be computed as aPAs (7-14)

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83 The scalar s is the distance from the given point P to the point A. Therefore it defines the size of the triangle. Since the size of the triangle does not matter while constructing neighborhood model, any value can be used for s. In this implementation 0.5 is used for s. After computing a, the vectors b can be computed by rotating a 120 degree about axis n and similarly c can be computed by rotating b 120 degrees. Then points B and C can be computed as. bPBs cPCs (7-15) When the number of surface normal vectors is two, the corresponding two triangles can be obtained by the following procedure. Let n1 and n2 be the two unit surface normal vectors at the point of interest. 11n and 12n ab Edge n1n2 cdACBDP Figure 7-7. Constructing two triangles from two normal vectors a is computed as the cross product of n 1 and n 2 21nna (7-16) b = -a c and d can be computed by cross product.

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84 anc1 bnd2 To use c and d vectors for constructing triangles, their direction should be verified. Since it is assumed that each solid primitive is the intersection of half spaces that are normal to n 1 and n 2 respectively, the following equations show its validity. If c = c. if not c = -c. 0cn2 If d = d. if not d = -d. 0dn1 After finding vectors a, b, c and d, the points composing vertices of the triangles can be computed as aPAs , C bPBs cPs and dPDs (7-17) Finally, when the number of surface normal vectors is three, the corresponding three triangles can be obtained by the following procedure. In Figure 7-8, n 1 n 2 and n 3 are unit normal vectors at the point of interest. 11n 12n and 13n Edge n1n3n2 ABPCa b cEdgeEdge Vertex Figure 7-8. Constructing three triangles from three normal vectors a is computed as the cross product between n 1 and n 3 a (7-18) 31nn If a = a. if not a = -a. 0an2 Similarly,

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85 b 12nn If b = b. if not b = -b. 0bn3 c 23nn If c = c. if not c = -c. 0cn1 After finding vectors a, b, c and d, the points composing the vertices of the triangles can be computed as aPAs and C bPBs cPs (7-19) After constructing the triangles for all the primitives whose boundaries pass through the point of interest, the neighborhood at the point is constructed by performing the Boolean operations between the triangles treating each triangle as a half space, which divides the space into two regions inside and outside. At first, intersections between triangles from the base and dependent solids are computed for each Boolean operation, and then the triangles will be subdivided at the intersections. Thereafter set membership classification of the triangles with respect to the half spaces (instead of the actual solid) is performed to collect the valid triangles. To classify the triangles, first the centroid Q of the triangle is computed and then this point is classified with respect to Boolean result recursively starting from the bottom of the CSG tree. It is assumed that the classification of the centroid is identical to the classification of the triangle. In addition, subdivided triangles that do not contain the point of interest are removed from the valid triangles because the neighborhood model can be constructed without those triangles. If there are triangles at the final node, it means that the set membership is on the solid. If there are no triangles, then it implies that the set membership is in or out. The above algorithm can be used for 2D or 3D CSG tree with the difference that lines are used to represent half spaces in 2D. Figure 7-9 shows an example that shows

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86 how to construct the neighborhood at a given point. As shown in Figure 7-9, the point of interest from A and B has multiple normal vectors. Two triangles and three triangles are derived from the solid A and B respectively. A union operation between those triangles is performed to get the valid triangles at the union node where four triangles are obtained after union. Therefore the point of interest can be classified as on the solid after union. AB A B U Figure 7-9. Neighborhood model using graphical method Figure 7-10 shows examples of solids created by Boolean operations using this software implementation

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87 Figure 7-10. Examples of 3D Boolean solids.

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CHAPTER 8 VOLUME OF THE SOLID MODEL 8.1 Linear Approximate Step Function Even though implicit surface equations can be used for representing the interior, exterior and boundary of the surface, it can not be directly used to compute the volume of the solid by integrating the equations because its values are not guaranteed to be 1 at the interior of the solid. Therefore an approximation using a linear step function is proposed here to construct a implicit function whose value is 1 in the interior of the solid. The linear approximate step function used is defined as follows. xxxxxh02211)( (8-1) Figure 8-1 Linear approximate step function graph x 10.5 h(x) Suppose f(x) is an implicit surface function where the region f(x) 0 is the solid, then the zero level set is the boundary of the solid. The zero level set function can be converted to the linear approximate step function by replacing x in Eq. 8-1 by the implicit 88

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89 function f(x) that is a scalar function. Then the set-membership classification at a given point can be performed by the linear approximate step function value as follows. ))((xfh > 0.5 inside the solid ))((xfh = 0.5 boundary of the solid (8-2) ))((xfh < 0.5 outside the solid The function provides an alternative implicit representation of the solid such that its value changes from 0 to 1 across the boundary of the solid. It will be therefore referred to as the function as the step function of the solid which can be used to define the solid as per Eq. 8-2. In Eq. 8-1, (())0.5xhf ))((xfh is a tunable parameter. As 0 the step function value becomes 1 almost everywhere in the interior of the solid. By using this property, the volume of the solid can be directly computed by integrating the step function of the solid. 21))((xfh0)( xf0)(xf0)( xf 1))((21xfh 21))((0xfh Figure 8-2. Transform from the implicit surface function to its linear approximate step function Boolean operation between two approximate step functions of the implicit functions f A and f B can be performed by the following equations by which a new step function representing the Boolean result can be constructed systematically.

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90 )()(ABBBAABAhhhhhhhhh Union operation )()(BABABABAhhhhhhhhh Intersection operation (8-3) )()(BABBAABAhhhhhhhhh Subtraction operation Where and h )(AAfhh )(BBfh These functions represent the step function of the Boolean between the solids represented by f A and f B The solid represented by this equation is not a regularized Boolean result but is suitable for computing the volume of the solid. The difference between ordinary Boolean result and the Regularized Boolean are lines in 2D or faces in 3d that do not contribute to the volume therefore the volume computed for ordinary Boolean is the same as the volume of the Regularized Boolean. Table 8-1. Simple examples for Boolean operation between approximate step functions where f 1 and f 2 both represent squares. BAh BAh BAh 1 BAh 21BAh 1BAh1BAh1 BAh1BAh 21BAh 0 BAh 21BAh 0BAh1BAh0 BAh0BAh 21BAh 1 BAh 21BAh 0BAh0BAh0 BAh1BAh 21BAh

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91 8.2 Constructing the Step Function for the CSG Solid A solid primitive is defined by intersections of the half spaces corresponding to the edges of the element and region where the density function has value greater than the level set value. The edges of the element can be defined by implicit functions defined in the parametric space. For example, in Figure 8-3, r =1 represents the right side of the element which divides the space into r>1 and r<1. The side where r =1 can be represented by the implicit function f 1 =1-r. Similarly all boundaries of the element can be represented by implicit functions. In Figure 8-3, all implicit functions within an element are displayed. 1rsf 11f 2f 3 f 4 f 5 rsf 1f 2f 3 f 6 t f 4f 5 f 7 (A) (B) Figure 8-3. Implicit surfaces defining a solid within an element in the parametric space A) 2D solid element B) 3D solid element In Figure 8-3A, the five implicit surface functions in a 2D solid element are f 1 = 1-r, f 2 = 1-s, f 3 = 1+r, f 4 = 1+s, f 5 = f = (r,s) b In Figure 8-3B, the seven implicit surface functions in a 3D solid element are f 1 = 1-r, f 2 = 1-s, f 3 = 1+r, f 4 = 1+s, f 5 = 1-t, f 6 = 1+t, f 7 = f = (r,s,t) b

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92 Since the Boolean operations between implicit functions in an element are all intersections, it can be simply treated as follows because it does not depend on the order of the Boolean operation. In a 2D element 5432154321ffffffefffffEhhhhhhhhh In a 3D element 76543217654321ffffffffefffffffEhhhhhhhhhhh (8-4) Where in 2D 4321ffffehhhhh 654321ffffffehhhhhhh in 3D Linear approximate step function representing the Boolean solid can be constructed by applying Boolean operation between solid elements whose step function is defined using Eq. 8-4. Hn Hn-1H1h1h0hn-1hn Figure 8-4. Labeling CSG tree nodes for constructing the approximate step-function Let the approximate step function for the Boolean solid be H n then it can be written as follows. ),(),(12111nnnnnnnhHBhhHBHH where n1 )(000fhhH (8-5)

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93 )(nnfhh In Eq. 8-5, new operators referred to as B 1 and B 2 are used. The definitions of B 1 and B 2 operators depend on the Boolean operation used. These definition are given in Table 8-2. Table 8-2. Function operator denoted by B where h A and h B are any arbitrary step functions Boolean type of F 1 U ),(1BAhhB )(BAhhh )(ABhhh )(BAhhh 2(,) A B B hh )(ABhhh )(BAhhh )(BAhhh As shown in Eq. 8-5, the linear step function for the solid can be constructed by expanding the equation recursively. It gives us a systematic way of constructing a single implicit function that represents the solid defined by the CSG tree. 8.3 Computing the volume of the solid The volume of the solid can be computed by integrating the approximate step function that represents the solid. Therefore the equation computing the volume of the solid can be simply written as dVHVnS (8-6) The domain of the integration is the interior of the solid. H n defined in Eq. 8-5 can be rewritten as follows. nnnChChChChH 221100 (8-7) The coefficients C n can be systematically constructed by the following equations. ),(),(),(),(113212111010nnhHBhHBhHBhhBC

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94 ),(),(),(),(112111112nnmmmmmmmhHBhHBhHBhHBC Where 0
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95 to integrate within this cell. If the subdivided element is completely inside, it is referred to as an internal cell and will be integrated by gauss quadrature. If the sudivided element is neither internal cell or external cell, it is a partial cell and that is subdivided again. This subdivision process is repeated until a termination criteria is satisfied. Therefore the total volume in an solid element is computed as the sum of the volumes of the internal cells and the partial cells. r s uv1111ith subdivided element (A) (B) (C) Node representing a internal cellNode representing a partial cellNode representing a external cell Node with descendant Figure 8-5 Subdivision of the 2D element and subdividing method A) Quadtree subdivision in a 2D element B) An example of quadtree representation structure C) An example of octree representation structure

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96 The isoparametric coordinate is used for integration as shown in Figure 8-5A. The integration within parametric space can be represented as the sum of the integration within cells. (a i c i ) is the left bottom corner point and (b i d i ) is the right upper corner point in the parametric coordinates at the i th cell in 2D. celliiiinnirsdcbannrsnnnnedrdssrCsrhdrdssrCsrhdVyxCyxhV11111det),(),(det),(),(),(),(JJ (8-10) [J] rs is the Jacobian matrix which is defined as follows. sysxryrxrsJ (8-11) A local coordinate system (u, v) for the subdivied cell as shown Figure 8-5A is used to integrate within the i th cell. Integration within the subdivided cell can be represented as dudvvuCvuhdrdssrCsrhVuvrsnnrsdcbanncelliiiiJJJdetdet),(),(det),(),(1111 (8-12) u, v is a local coordinate system defined for the subdivided element. The relation between r, s coordinate to the u, v coordinate is iiiauabr)1(2)( 2)(iiabur 0 vr iiicvcds)1(2)( 0 ur 2)(iicdvs (8-13)

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97 Jacobian matrix [J] uv can be computed as 2)(002)(iiiiuvcdabvsvrusurJ (8-14) Similarly the following equations are obtained for 3D. (a i c i e i )is the left bottom corner point at the front face and (b i d i f i )is the right upper corner point at the back face in the parametric coordinates at the i th cell in 3D. celliiiiiinnifedcbarstnnrstnnnnedrdsdttsrCtsrhdrdsdttsrCtsrhdVzyxCzyxhV1111111det),,(),,(det),,(),,(),,(),,(JJ (8-15) dudvdwwvuCwvuhdrdsdttsrCtsrhVuvwrstnnfedcbarstnncelliiiiiiJJJdetdet),,(),,(det),,(),,(111111 (8-16) [J] rst is the Jacobian matrix defined as wzwywxvzvyvxuzuyuxrstJ (8-17) u, v, w are the local coordinates of the subdivided cell. The relation between r, s, t coordinate and the u, v w coordinate is iiiauabr)1(2)( 2)(iiabur 0 vr 0 wr

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98 iiicvcds)1(2)( 0 us 2)(iicdvs 0 ws iiieweft)1(2)( 0 ut 0 vt 2)(iiefwt Jacobian matrix [J] uvw can be computed as ()002()02()002 0 Jiiiiuvwiibarstuuudcrstvvv f erstwww (8-18) Figure 8-6 shows the union between square and circle that is used for comparing the numerical integration results by changing some parameters such as subdivision number, order of integration and 1 1 R0.5 AB BA 589.15.043112Area Figure 8-6. A simple 2D example used for comparing the numerical integration results by changing some parameters such as subdivision number, order of integration and Table 8-3. Volume integration of Figure 8-6 by changing subdivision number where exact volume is 1.589 (Order of integration = 3, =0.001) Subdivision # 1 2 3 4 5 6 7 1.50694 1.59432 1.59852 1.58857 1.58955 1.58876 1.58890 Error (%) -5.164 0.334 0.599 -0.027 0.034 -0.015 -0.006

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99 Table 8-4. Volume integration of Figure 8-6 by changing the order of gauss integration where exact volume is 1.589 (Subdivision # = 4, =0.001) Order of Integration 1 2 3 4 5 1.60937 1.59472 1.58857 1.58831 1.59019 Error (%) 1.281 0.359 -0.027 -0.043 0.074 Table 8-5. Volume integration of Figure 8-6 by changing in the linear step function where exact volume is 1.589 (Order of integration = 3, Subdivision # = 4) 1 0.1 0.01 0.001 0.0001 0.92778 1.55700 1.58874 1.58857 1.58857 Error (%) -41.612 -2.013 -0.016 -0.027 -0.027 Table 8-3 shows that a more accurate value by increasing the number of subdivisions can be obtained. Table 8-4 shows that 3 rd order integration is sufficient to compute the volume. In other words, increasing the order of the integration in gauss quadrature further does not give us better solution. Finally, in Table 8-5, it is known that large value for gives us an incorrect answer. The following tables show more examples of computing the volume of solids. Table 8-6. Examples for computing the volume of the solid 2D Primitives Exact Volume 1.0 0.78477 1.57330 0.78628 0.21371 Integration 1.00000 0.78539 1.57079 0.78539 0.21460 Error (%) 0.000 0.079 -0.159 -0.113 0.416 Boolean 2D Exact Volume 1.589 1.78539 0.80365 0.71725 Integration 1.58857 1.78477 0.80380 0.71838 Error (%) -0.027 -0.034 0.018 0.157

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100 3D Examples Exact Volume 1.0 3.14159 0.58333 1.589 0.35862 Integration 1.00000 3.14159 0.58333 1.59852 0.35964 Error (%) 0.000 0.000 0.000 0.599 0.284 So far it has been only investigated the method to compute the volume of the solid has been investigated. Surface area is another volumetric properties of the solid that needs to be computed. One way of computing the surface areas is by summation of the areas of triangles generated for graphical display of the solid. To compute the surface area of the solid more systematically further research is needed.

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CHAPTER 9 ALGORITHM FOR GRAPHICAL DISPLAY To display the solid graphically it is necessary to polygonize or tessellate the surface of the solid. Many algorithms have been developed [Bloomenthal, 1994], [Wyvill and Overveld, 1997] for algebraic surfaces that can be easily adapted to generate graphical display of primitives represented using implicit solid elements. An algorithm has been developed that first tessellates the primitives and then uses these surface triangles to construct the graphical display for the solid created by Boolean combination of these primitives. The intersections between surface-triangles from different primitives are computed followed by set membership classification to eliminate unnecessary triangles. The details of this graphics algorithm are discussed in the following sections. 9.1 Overview of Computer Graphics Even though there are the exact equations for curves in 2D and surfaces in 3D, their computer graphics is displayed on the screen using lines and polygonal approximations. Figure 9-1. Graphics of a curve As shown in Figure 9-1, the graphics of a curve is composed of many line segments where the number of line segments composing the curve determines the resolution of the graphics. In other words, more line segments are required to obtain better graphics of the 101

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102 curve. Such lines represented by two points need to be computed and stored for displaying 2D graphics. Since this program is implemented by Java and Java3D, such lines representing 2D graphics are stored in a class called LineArray. Figure 9-2. Graphics of a sphere and its triangulation Graphics for surfaces can be represented by any type of polygons such as triangles or quadrilaterals. Figure 9-2 shows an example where a sphere is represented by many triangles. As shown in Figure 9-2, the graphics of a surface is composed of many triangles where the number of triangles composing the surface determines the resolution of the graphics. The triangles are defined by three vertex points that need to be computed and stored as arrays for displaying its 3D graphics. A class called TriangleArray is used in Java3D for storing a triangulated approximation of surfaces for graphics display. 9.2 2D Primitive Display Points on the boundary of the solid can be computed as intersection of rays from the center of the element in general. A ray from the center in the direction can be expressed as in the parametric space. The vectors n rsnnin j unu)(R r and n s are expressed as cosine and sine of angle varying from 0 to 2 to evenly divide the space.

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103 The intersection of this ray with the surface can be computed by solving the equation rsb(nu,nu)0 (9-1) Since the above equation is nonlinear, it has to be solved using Newton-Raphson method in general (Point A in Figure 9-3). Note that the intersection of the ray and boundary of the solid must be within the element. If no intersection of the ray and the implicit Eq. 9-1 is found within the element then the intersection of the ray with the boundary of the element needs to be found (Point B in Figure 9-3). The following figure shows an example in the parametric space. rs11 bsr ),()(uRA )(uRB Figure 9-3 Ray methods for finding points on the boundary of the solid in 2D The LineArray for displaying the boundary of the solid in 2D is constructed by connecting points computed by ray intersection with the boundary. Let a point on the boundary of the solid be P() which satisfies Eq. (9-1). Then the neighboring point can be represented by P(+) where is the step size of the angle increment. The resolution of the 2D graphics depends on Therefore, smaller must be used to obtain higher resolution for the graphics of the 2D solid.

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104 When the solid primitive to be displayed has an analytic equation, points on the boundary can be found faster by directly using the analytic (parametric) equation of the boundary. For quadrilaterals, the nodes of the element can be connected to form the required line array. For an ellipse, the parametric coordinates of points on the boundary of the ellipse can be easily computed using the equation )sin,(cos),(**sr Once the parametric coordinates of a point on the boundary are found, it can be mapped to the global coordinates by the mapping function X(r, s). ),(**sr xyrs11 )sin,(cos),(**sr )sin,(cos),(**XXsr Figure 9-4. Computing the points on the boundary of the ellipse Even though all lines composing a 2D primitive can be saved in one LineArray, they are separately listed in separate LineArrays where each LineArray represents each edge in a 2D primitive. This is done for simplifying the algorithm to compute the graphics for the Boolean result as described later in Section 9.5. 9.3 2D Boolean Result Display At first in this algorithm for computing the LineArray to display the result of a 2D Boolean operation, all the intersection points between the line segments of the base and dependent solids are computed first. Thereafter the lines having intersections are divided at these intersection points. This is based on the idea that the resultant geometry is

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105 composed of lines divided at the intersections. Finally, set-membership classification is applied to each divided line segment to determine line segments that are on the surface of the Boolean result. P (Intersection point)B_v2B_v3 B_v4 D_v1D_v2D_v3 D_v4B_v1 B_v2B_v1B_v2B_v3B_v3B_v4B_v4PPB_v1 D_v2D_v1D_v2D_v3D_v3D_v4D_v4PPD_v1 Union Dependent solidBase solid Figure 9-5. Union of two squares in 2D As shown in Figure 9-5, the side B_v1-B_v4 in the base solid and the side D_v1-D_v4 in the dependent solid are divided by the intersection point P. Therefore five line segments each are obtained for the base and dependent solid. After applying set membership classification on all line segments, the union result from the valid six line segments is constructed. The detailed procedure for 2D Boolean operation is explained in the following. Input: n LineArrays from the base solid m LineArrays from the dependent solid Output: LineArrays composing the Boolean solid 1. for i = 1 to n 2. if i th LineArray from the base solid intersects the bounding box of the dependent 3 solid element 4 then 5 for j =1 to m 6 for k = 1to number of lines composing i th LineArray 7 for l = 1 to number of lines composing j th LineArray 8 find intersection between two lines and save the intersection point 8 } // end of l 9 } // end of k 10 } // end of j 12 else do set-membership classification on any line composing i th LineArray 11 if on the solid 12 then add i th LineArray for the Boolean solid

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106 13 else discard i th LineArray 14 divide lines at the intersection points for i th LineArray 16 perform set-membership classification on each line 17 collect valid lines and make new LineArrays 18 } // end of i 19 for j =1 to m 20 divide lines at intersection points for j th LineArray of the dependent solid 21 perform set-membership classification on each line segment of the array 22 collect valid lines and make new LineArrays 23 } // end of j As written in line 2 above, the bounding box method is used to check if an element intersects another element. The bounding boxes are constructed by simply creating rectangles by using the maximum and the minimum values in the nodal coordinates as shown in the following figure. solid elementBounding box of the solidelement xy Figure 9-6. Bounding box for a 2D solid element Note that bounding box method only tells us that there is no intersection between two solid elements when bounding boxes do not intersect each other. In other words, two solid elements may not intersect each other even though their bounding boxes intersect each other. 9.4 3D Primitive Display There are two different algorithms for creating 3D primitive graphics depending on whether the cross-section is constant or varying. For constant cross-sectional sweep, the

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107 cross-section does not vary in the t direction. Therefore, points X(r s -1) on the boundary of the 2D profile to be swept can be computed first and then similar points for other t values as X(r s -1+jt) can be computed. rs11 ),(**srP ),,(**tsrX ),,(**tsrX ),,(**tsrX (A)(B)(C)(D) Figure 9-7. Computing the points on the boundary of the solid A) 2D profile to be swept B) Points on the extrude C) Points on the revolve D) Points on the sweep As mentioned in section 9-1, TriangleArray is used for saving triangles where the number of TriangleArrays for representing a 3D primitive depends on the number of LineArray composing the 2D profile used. For varying cross-sectional sweep, a more general method for finding points on the boundary of the solid is needed. The points on the boundary of the solid can be computed as the intersection of rays from the center of the cross-section. A ray from the center in the direction can be expressed as rsnnni j uunR )( in the parametric space. Similar

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108 to 2D, n r and n s are expressed as cosine and sine of angle varying from 0 to 2 to evenly divide the space. The intersection of this ray with the surface can be computed by solving the equation 0),,(*bsrtunun (9-2) where, t* is the value of t at which points on the boundary are computed using this ray algorithm. Since the above equation is nonlinear, it has to be solved using the Newton-Raphson method in general. Cross section where t = t*t=t* r11 )(uR Figure 9-8. Finding the points on the boundary of the varying cross section sweep After finding such points on the boundary of the solid for many evenly spaced points in the t direction, these points are connected to construct the triangles needed to display the 3D primitive solid. 9.5 3D Boolean Result Display The triangle array to display the 3D Boolean result between the base and the dependent solid is constructed directly from the triangle arrays of the base and dependent solid. The elements of both base and dependent solid are first classified into intersecting or non-intersecting elements so that only triangles of the intersecting elements need to be investigated to find intersecting triangles thus simplifying the computation. After

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109 computing all intersection between triangles to compute intersection points and lines, the intersected triangles are sub-divided into smaller triangles by re-triangulating the vertices and intersection points. Finally, set-membership classification is performed to classify all triangles so as to collect all valid triangles that represent the Boolean result. The procedure for computing the triangle array for the 3D Boolean result can be stated as the following four steps. Step1) Determine element to element intersection Step2) Compute intersection points and lines from triangle to triangle intersection Step3) Re-triangulate intersected triangles to include intersection points and lines Step4) Perform set-membership classification to collect valid triangles These steps are described in more detail below. Step1) Determine element-element intersection First of all, elements from base and dependent solids that intersect are determined to minimize the total number of triangles that need to be searched to find intersections. To obtain the benefit from this procedure, each triangle should be labeled so that the 3D elements that contributed to creating the triangle can be determined. Before any Boolean operations are performed each triangle is associated with a unique element. However, during the Boolean operations if faces of two elements are coincident then after the Boolean all the triangles on the coincident face are labeled as belonging to both the elements. For each Boolean operation, only a few elements intersect each other, even though the solids are composed of a lot of elements. Identifying the intersecting elements will dramatically decrease the number of triangles that have to be search for possible

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110 intersection. All 3D elements of the dependent solid are compared with all 3D elements of the base solid in order to find intersecting elements. Boolean solid Dependent solid Base solid N on-Intersected element node Intersected element node Figure 9-9. 3D CSG tree structure Figure 9-9 shows a solid represented by a CSG tree where each 3D primitive node is another binary tree. In other words, each 3D primitive is composed of one or more 3D elements. Although the number of triangles for each solid is very large, the number of triangles associated with the intersecting elements is much smaller. All TriangleArrays associated with the intersecting elements are collected for the next step. Then set-membership classification for non-intersecting triangle arrays is performed to determine valid TriangleArrays among all non-intersecting TriangleArrays. If one of the triangles from a TriangleArray is on the Boolean solid, the entire TriangleArray must be part of the Boolean solid. Figure 9-9 illustrates the TriangleArrays of the base and dependent solid as bars where each segment of the bar represents a triangle. The total number of TriangleArrays of the base and the dependent solid are N and M respectively. M de p endent Trian g leArra y s N base Trian g leArra y s

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111 Figure 9-10. All triangleArrays composing the base and dependent solid To check whether two elements intersect bounding cylinder method is used. The sweep trajectories connecting the centroids of the profile elements are defined for the two elements of interest where sweep trajectory can be line in extrude, arc in revolve or arbitrary curve in sweep. Thereafter, the closest points between the two trajectories are computed as the optimal solutions obtained by minimizing )()(21ttSS as shown in Figure 9-10. Let D1 be the maximum diagonal at the 2D profiles used for creating element 1and D2 be the maximum diagonal at the 2D profiles used for creating element 2. Then the minimum distance between two trajectories is compared with D1+D2 to determine whether the bounding cylinders intersect. Let the minimum distance between two trajectories be d, then If d > D1+D2 bounding cylinders do not intersect If d D1+D2 bounding cylinders intersect Since bounding cylinder method is just an approximation, the bounding cylinder method only tells us that there is no intersection between two solid elements when the bounding cylinders do not intersect each other. In other words, two solid elements may not intersect each other even though their bounding cylinders intersect each other.

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112 Sweep trajectory(S1(t)) Closest points betweentwo trajectories D1 D2Sweep trajectory(S2(t)) Figure 9-11. Bounding cylinder method between two elements Suppose the number of TriangleArrays from the intersected elements is n for the base and m for the dependent, N-n TriangleArrays from the base and M-m TriangleArrays from the dependent are either added to or removed from the Boolean solid by the set-membership classification. Therefore, only n TriangleArrays from the base solid and m TriangleArrays from the dependent solid will be considered in the second step. n base Trian g leArra y s m de p endent Trian g leA r ra y s Figure 9-12. All triangleArrays relevant to intersected elements Step 2) Searching for intersection points and lines on triangles As stated earlier, intersection between triangles that belong to the intersecting elements is searched for now. Intersecting triangles may be coincident or intersecting along a line. If two triangles are coincident, then intersection between edges of the

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113 triangles to find Edge-Edge intersection points is searched for. On the other hand if the triangles are intersecting but not coincident, then intersection between the edges of each triangle with the other triangle is searched for to find Plane-Edge intersection point. Representative examples of these two cases are shown in the following figure. ( B ) ( A ) Figure 9-13. Triangle-Triangle intersection A) Plane-Edge intersection B) Edge-Edge intersection For each triangle, intersection points with edges of other triangles and intersection lines with other triangles need to be stored so that they can be used for the next step of the algorithm. Suppose the number of TriangleArrays having actually intersected triangles is n for the base and m for the dependent, n-n non-intersecting TriangleArrays from the base and m-m non-intersecting TriangleArrays from the dependent are either added to or removed from the graphics by performing set-membership classification. Therefore, only n TriangleArrays from the base solid and m TriangleArrays from the dependent solid will be considered in the third step. n base TriangleArrays m dependent TriangleArrays N on-intersected trian g le Intersected trian g le Figure 9-14. TriangleArrays having actually intersected triangles

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114 Step3) Triangulation Different types of triangulation procedures are used to handle the coincident and intersecting triangles. If the two or more triangles are coincident, all these triangles will be simultaneously triangulated. On the other hand intersecting triangle are triangulated individually. The well known Delaunay triangulation algorithm with some minor modifications is used in this system. Since Delaunay triangulation algorithm works for points on a plane, a mapping procedure to transform points in 3D to points in 2D is needed. This mapping is described below. P P u w v V3 V1 V2 V2 V3 V1 v y z u x Figure 9-15. Mapping the triangle in space onto the local u-v plane In Figure 9-14, u is a unit vector connecting V 1 to V 2 w is the unit vector which is normal to the triangle. Then v can be computed by v=wu Let a point P be a given point on the triangle. Its coordinates (u, v) with respect to a local 2D coordinates system is computed as u = d cos v = d sin where d = distance from the V 1 to P = angle between u and the vector V 1 -P Once all points are mapped to u-v plane, those points are used for the Delaunay triangulation. The swap procedure in the triangulation algorithm is modified to make sure

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115 that the intersection lines included as edges in the resulting triangulation. In other words, the intersection lines saved in the triangle are not swapped during Delaunay triangulation so that these lines will be included as edges of the triangles. During the triangulation, one triangle may be divided into two or more triangles as shown in Figure 9-16. Figure 9-16. Subdivided triangles after Delaunay triangulation Step4) Set-membership classification It can now be distinguished between valid triangles and invalid triangles by the set-membership of the centroid of the triangle. In other words, if the centroid of a triangle is outside or inside the solid, the triangle will be removed from the TriangleArrays representing the Boolean result otherwise it is included. In this final step, all TriangleArrays having intersected triangles are destroyed and new TriangleArrays are created. The following is the flow chart for these steps.

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116 Task 3: Task 2: Task 1: Triangulation for each triangle Triangulation for all coincident p lanes simultaneously Set-membership classification on N -n non-intersected base TriangleArrays M-m non-intersected base TriangleArrays Collect valid TriangleArrays and add to the Boolean solid End Set-membership classification Collect valid triangles and create new TriangleArrays which will be added to the Boolean solid Finding intersections on all intersected triangles Element-Element Intersection n intersected base TriangleArrays m intersected dependent TriangleArrays N o Yes N base TriangleArrays M dependent TriangleArrays Start Coincident cases exist Task 4: Figure 9-17. Flow chart for constructing the graphics for the 3D Boolean result 9.6 Discussions and Suggestion for improvement The method used for displaying the graphics of the solids in this system is not very efficient and needs to be improved to obtain better display speed. There are many reasons for the inefficiency of the algorithm. Currently an iterative numerical method is used find points on the boundary of the primitives. This cannot be avoided if the equation representing the intersection of a ray and the boundary is a high degree nonlinear equation. Using analytical solution whenever possible instead of numerical method, can significantly improve the graphics display

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117 speed. Set membership classification needs to be performed repeatedly during the procedure to classify a very large number of triangles. Better algorithms for determining element to element intersection, element to triangle array intersection or triangle to element intersection can reduce the number of triangles whose intersections need to be computed.

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CHAPTER 10 CONCLUSION AND DISCUSSION 10.1 Conclusion In this study, an approach for representing primitive solids using implicit solid elements is presented. These primitives can be combined using Boolean operation to create more complex solids. In traditional CSG approach the primitives are defined by Boolean combination of half-spaces. Instead, in this approach, the primitives are defined using implicit surface functions that are represented parametrically within quadrilateral or hexahedral elements. By representing the density function parametrically, many of the advantages of parametric curve and surface representation such as axes independence and ease of tracing are inherited by the implicit solid element representation. A primitive represented using implicit solid element can be edited by changing the nodal density values, level set value ( b ) or by changing the size of the element. The implicit solid elements introduced in this study enable the representation of sweep solids using implicit surface boundaries. The solid elements for representing sweep solids are classified by the type of sweep trajectory as well as by the cross-section shapes. Implicit solid elements were developed for each type of sweep solid including extrude, revolve and general sweep. In general the biggest draw back of implicit surface representation is that it is necessary to solve equations to compute points on the boundaries of the solid. However, when the implicit surface functions are defined within quadrilateral and hexahedral elements that are not distorted, fairly simple algorithms can be developed for generating tesellations required for graphical display. The main 118

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119 advantage is that set membership classification is easier for implicit solid elements compared to B-Rep models especially when the elements are not distorted. Another advantage is that Boundary evaluation algorithms are not required when the solid is represented using either implicit half spaces (as in conventional CSG technique) or when implicit solid elements are used to define the primitives. Furthermore when the implicit solid elements are used for representing the solids additional information such as material properties can be super-impose by assigning nodal values for these properties. The variation of these properties within the solid can be represented using appropriate interpolation within the element. If necessary, the element can be sub-divided into smaller elements to provide the necessary resolution to represent material variation required to represent any type of heterogeneous solids. The algorithms presented in this study are implemented using Java and Java3D to verify the algorithms and to compare performance with B-Rep models. 10.2 Future Work Since this approach is just introduced, there are still many problems to be solved. It is required to develop implicit solid elements for freeform modeling as well as for representing other types of features routinely used in current CAD systems. And most solid models can be used for FEA (Finite Element Analysis) after generating meshes on them and for finding NC tool path for manufacturing. To establish such abilities in this solid modeling system, further study is needed. Without developing these techniques implicit solid modeling is unlikely to be a viable alternative to the currently popular B-Rep technology.

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APPENDIX SCENEGRAPH AND CLASS STRUCTURE A.1 SceneGraph Structure for Java3D Locale Base SolidDependent SolidBranchGroup(BG)TransformGroup(TG)BGFaceEdgeVertex MeshNodeFlipArrowDimensionLineDimensionRefEdgeRefFaceRefVertexBGBGTG PickTranslatePickRotatePickZoom ViewModel MousePick Figure A-1 SceneGraph structure for the solid modeler using implicit solid elements 120

PAGE 135

121 A.2 SDModeler Classes Class structure Packages commandList constraint delauny equationSolver graphicsComponent interactiveCommand shapedensity solidModel userInterface Main Class : SDModeler commandList ClAssembly ClBoolean ClEdit ClFile ClHelp ClMain ClSelect ClSolid2D ClSolid3D ClView delauney Dtriangle constraint ConstraintSet PositionConstraint Utility VL_C VL_D VP_D VV_A VV_C VV_D VV_P graphicsComponent Curve CurveHermite DimensionText2D Line PickPoint PickPtElementNode PickPtIntersection PickPtstatic PickVector SdLineArray SdPointArray SdTriangleArray Triangle UtilGeometry equationSolver CCNewtonRaphson CLNewtonRaphson EquationSolver InteractiveCommand IcAlignLines IcDimensionChange IcExtrude IcRevolve IcSolid2DPrimitives IcVLC IcVLD IcVPD IcVVA IcVVC IcVVD IcVVP InteractiveCommand shapedensity Q4Node Q8Node Q9Node Sd3DExtrude Sd3DRevolve Sd3DSMBlend Sd3DSTBlend Sd3DSweep ShapeDensity ShapeDensity2D ShapeDensity3D T10Node solidModel CSG_History Constraints CsgNode Solid Solid2D Solid2dCircle Solid2dEllipse Solid2dRectangle Solid2dTriangle Solid2dSemiCircle Solid2dQuaterCircle Solid2dWedge Solid3D Solid3dExtrude Solid3dRevolve Solid3dSMBlend Solid3dSTBlend Solid3dSweep userInterface FileIO Java3DUniverse MainDialogBoxes MousePick SDModelerFrame ViewCanvas

PAGE 136

122 A.3 SDModeler Class Structure ViewCanvas Solid CSG_History CsgNode Java3DUniverse SDModelerFrame CommandList JFrame SDModeler Canvas3D Constraints ShapeDensity Solid2D Solid3D Solid2dCircle Solid2dEllipse Solid2dRectangle Solid3dExtrude Solid3dSweep InteractiveCommand Solid3dRevolve Solid3dSweepSTBlend Solid3dSweepSMBlend Solid3dSTBlend Solid3dSMBlend Solid2dSemiCircle Solid2dQuaterCircle Solid2dTriangle Solid2dWedge Figure A-2 Class structure for the solid modeler using implicit solid elements

PAGE 137

123 ShapeDensity2D ShapeDensity ShapeDensity3D Q4Node Q8Node Q9Node Sd3DExtrude Sd3DRevolve Sd3DSweep Sd3DSTBlend Sd3DSMBlend Sd3DSweepSTBlend Sd3DSweepSMBlend Figure A-3 ShapeDensity class structure InteractiveCommand IcExtrude IcVLc IcVLD IcVPD IcVVA IcVVC IcVVD IcVVP IcDimensionChange IcRevolve IcSolid2DPrimitives IcAlignLines Figure A-4 InteractiveCommand class structure

PAGE 138

124 CommandList ClSolid3D ClAssembly ClBoolean ClEdit ClFile ClHelp ClMain ClSelect ClSollid2D ClView Figure A-5 CommandList class structure ConstraintSet VP_D VP_D VP_D VP_D VP_D VL_D VL_C PositionConstraint Figure A-6 PositionConstraint class structure

PAGE 139

LIST OF REFERENCES Bajaj C, Chandrajit L, Chen J and Xu G, 1995, Modeling with Cubic A-Patches, ACM Transactions on Graphics, 14(2), pp.103-133. Bloomenthal, J., 1994, An Implicit Surface Polygonizer, Graphics Gems IV, New York, Academic Press. Bloomenthal J, Bajaj C, Blinn J, Cani-Gascuel MP, Rockwood A, Wyvill B and Wywill G, 1997, Introduction to Implicit Surfaces, Morgan Kaufmann Publisher Inc, San Francisco, CA. Choi B.K. and Lee C.S., 1990, Sweep Surfaces Modeling via Coordinate Transformations and Blending. Computer-Aided Design, 22:87-96. Filip, D. J.and Ball, T. W., 1989, Procedurally Representing Lofted surfaces, IEEE Computer Graphics and Applications v 9 n6 p 27-33. Glassener A, 1989, An Introduction to Ray Tracing, Academic Press, London. Hoffmann C, 1987, Geometric and Solid Modeling-An Introduction, Morgan Kaufmann Publishers, San Mateo, CA. Kumar, A. V. and Gossard D. C., 1996, Synthesis of Optimal Shape and Topology of Structures, Journal of Mechanical Design, Transaction of the ASME, vol. 118, no. 1, pp. 68. Kumar, A. V. and Yu, L., 2001, Sequential Constraint Imposition for Dimension-Driven Solid Models, Computer Aided Design, 33(6), pp. 475-486. Lee K, 1999, Principles of CAD/CAM/CAE System, Addison Wesley, Reading, MA. Lorensen W. and Cline H., 1987, Marching Cubes-A High Resolution 3D Surface Construction Algorithm, Computer Graphics, 21(4), pp.163-69. Mantyla M, 1988, An Introduction of Solid Modeling, Computer Science Press, Rockville, MD. Tigges M. and Wyvill B., 1999, A Field Interpolated Texture Mapping Algorithm for Skeletal Implicit Surfaces, Proceedings of Computer Graphics International, Canmore, Canada. 125

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126 Martin R.R. and Stephenson, 1990, P.C. Sweeping of Three-Dimensional Objects. Computer-Aided Design 22:223-234. Mortenson ME, 1997, Geometric Modeling Second Edition, Wiley Computer Publishing, John Wiley and Sons, New York. Osher S. and Redkiw R., 2003, Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag Inc., New York Requicha, AAG and Voelcker HB, 1985, Boolean Operations in Solid Modeling: Boundary Evaluation and Merging Algorithms, Proceedings of the IEEE, 73(2), pp. 30-40. Shapiro V., 1998, Theory of R-functions and Applications: A Primer. CPA Technical Report CPA88-3, Cornell Programmable Automation, Sibley School of Mechanical Engineering, Ithaca, NY 14853. Shapiro V. and Tsukanov, 1999, Meshfree Simulation of Deforming Domains, Computer-Aided Design, 31, pp. 459-471. Spatial Technology Inc., 1995, Geometry Modeler Application Guide, Spatial Technology Inc., Boulder, CO. Wyvill, B. and Overveld, K. V., 1997, Polygonization of Implicit Surfaces with Constructive Solid Geometry, J. of Shape Modeling, 2(4), pp. 257-273. Zeid I.,1991, CAD/CAM Theory and Practice. McG Computer Science Series, Singapore. Zhao HK, Osher S, Merriman B and Kang M, 2000, Implicit and NonParametric Shape Reconstruction from Unorganized Data Using a Variational Level Set Method, Computer Vision and Image Understanding, 80, pp.295-314

PAGE 141

BIOGRAPHICAL SKETCH The author was born on September 6, 1970, in Seoul, Korea. In 1996, he graduated with a Bachelor of Science degree in mechanical engineering from Inha University, Inchon, Korea. He entered the Master of Science program in the Mechanical Engineering Department at the University of Florida in 1997 and obtained his masters degree in 1998. Thereafter he continued to study in the Ph.D. program. On November 23, 1996, he was married to Duyoung Kim. He is the father of two lovely daughters, Jasmine and Anna. 127


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SOLID MODELING USING IMPLICIT SOLID ELEMENTS


By

JONGHO LEE













A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2003

































Copyright 2003

by

Jongho Lee

































To my lovely family















ACKNOWLEDGMENTS

I would like to express my sincere gratitude to my advisor and the chairman of my

supervisory committee, Dr. Ashok V. Kumar, for his guidance, encouragement and

patience throughout my study. I feel myself very fortunate not only to have benefited

from his academic guidance but also to have enjoyed his invaluable friendship. Without

his assistance this study would never have been completed.

I am also grateful to my committee members, Dr. John C. Ziegert, Dr. John K.

Schueller, Dr. Carl C. Crane and Dr. Baba C. Vemuri, for their advice, comments and

patience in reviewing this dissertation.

I would also like to thank my parents, brothers, my wife, Duyoung Kim, and my

lovely two daughters, Jasmine and Anna, for their valuable love and continuous support

during my study.
















TABLE OF CONTENTS
Page

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TABLES .............................. ..... .... .. .. ............... ....... viii

LIST OF FIGURES ......... ......................... ...... ........ ............ ix

A B S T R A C T .......................................... ..................................................x iii

CHAPTER

1 IN TR OD U CTION ............................................... .. ......................... ..

1.1 O overview ..................................................................................................... ....... 1
1.2 G oal and O bjectives.......... ......................................................... ............... .4
1.3 O utlin e ......................................................................... . 4

2 SOLID REPRESENTATION SCHEMES ....................................... ............... 6

2 .1 G eom etric M o d elin g .................................................................... .....................6
2.2 Constructive Solid Geometry (CSG).............................................. ............... 7
2.3 Boundary Representation (B-Rep) ............................................... ............... 11
2.4 H ybrid System ............................................ .. .. .... ........ ......... 13
2.5 Sw eep Features ........................ .... .. .. .................. .......... .... ............... 14
2.6 Im plicit Surfaces or Level Set Surfaces..................................... ............... 15

3 IM PLICIT SOLID ELEM EN TS ........................................ .......................... 17

3.1 Definition of Implicit Solid Elements...........................................................17
3.2 Definition of the Solid within an Implicit Solid Element............... ................ 20
3.3 3D H exahedral Solid Elem ents....................................... .......................... 21
3.4 D efining and Editing Prim itives .......................... .... ......... ............... .... 23
3.5 Defining Complex Geometries using Implicit Solid Elements ..........................25
3.6 Heterogeneous Solid Model Capability.....................................................26

4 TWO DIMENSIONAL PRIMITIVES ............................................ ............... 28

4.1 2D Primitives by 9-Node Quadrilateral Elements..............................................28
4.2 M apping in 2D Solid Elem ents....................................... .......................... 32









5 CONSTANT CROSS-SECTION SWEEP......................................................36

5.1 E xtrude E lem ents ................................................................ .. .. .............. 37
5.2 Revolve Elements .................................. .. .. .. ........ ...............40
5.3 Sw eep Elem ents .................. ...................................... .. ...... .... 42

6 VARIABLE CROSS SECTION SWEEP ...................................... ............... 46

6 .1 Straight B lend E lem ent.............................................................. .....................47
6.2 Sm ooth Blend Elem ent........................................................... ............... 55
6 .3 S w e ep B len d ................................................................. ................................6 7

7 CSG REPRESENTATION USING IMPLICIT ELEMENTS .................................69

7.1 Surface N orm al V ectors .......................................................... .....................69
7.2 Set membership classification within an element............... ............ 74
7.3 Constructive Solid Geometry using Implicit Solid Elements...............................77

8 VOLUME OF THE SOLID MODEL ............................................. ............... 88

8.1 Linear Approximate Step Function .................................................. 88
8.2 Constructing the Step Function for the CSG Solid................................... 91
8.3 Computing the volume of the solid .............................. ...............93

9 ALGORITHM FOR GRAPHICAL DISPLAY................................... .................. 101

9.1 Overview of Com puter Graphics............................................... .................. 101
9.2 2D Prim itive D display .......................................................... ............... 102
9.3 2D B oolean R esult D display ........................................ .......................... 104
9.4 3D Prim itive D display .......................................................... ............... 106
9.5 3D B oolean R esult D display ........................................ .......................... 108
9.6 Discussions and Suggestion for improvement..............................116

10 CONCLU SION AND DISCUSSION ................................................................. 118

10.1 Conclusion .................................... ............................... ......... 118
10.2 Future W ork .................. .................. ................ .......... .......... .... 119

APPENDIX

SCENEGRAPH AND CLASS STRUCTURE..........................................................120

A. 1 SceneGraph Structure for Java3D............... ...................... ............... 120
A .2 SD M odeler C lasses................................................. .............................. 121
A .3 SD M odeler Class Structure ........................................ ........................ 122









L IST O F R E FE R E N C E S ......................................................................... ................... 125

BIOGRAPHICAL SKETCH ............................................................. ..................127
















LIST OF TABLES


Table pge

3-1. Basis functions for the 9 node quadrilateral element .............................................. 19

3-2. Hexahedral 8-node basis functions....... .. ........... .... ..................... ........... 22

3-3. Hexahedral 18-node basis functions................ ......................... ...............22

4-1. Samples of 2D primitive design ......... ....... ....................................... 30

4-2. Analytical solutions for the 4 node quadratic element mapping (parallelogram) ......34

4-3. Analytical solutions for the 4 node quadratic element mapping (non-parallelogram)35

7-1. Surface normal vectors on the boundary of the element ..........................................73

8-1. Simple examples for Boolean operation between approximate step functions where fi
and f2 both represent squares......................................................... ............... 90

8-2. Function operator denoted by B where hA and hB are any arbitrary step functions ....93

8-3. Volume integration of Figure 8-6 by changing subdivision number where exact
volume is 1.589 (Order of integration = 3, e =0.001)...................................98

8-4. Volume integration of Figure 8-6 by changing the order of gauss integration where
exact volume is 1.589 (Subdivision # = 4, e =0.001) .......................................... 99

8-5. Volume integration of Figure 8-6 by changing E in the linear step function where
exact volume is 1.589 (Order of integration = 3, Subdivision # = 4)....................99

8-6. Examples for computing the volume of the solid .............................................99
















LIST OF FIGURES


Figure pge

2-1. Exam ple of C SG binary tree structure................................... ...................... ........... 8

2-2. Comparison between the ordinary Boolean and regularized Boolean operation .........8

2-3. Regularized Boolean treatment in overlapped case.....................................................9

2-4. Examples of neighborhood models on the vertices in 2D.............................. 10

2-5. Mathematical description of the primitives by half-spaces A) Half-space by x-y
plane B) Block by half spaces C) Cylinder by half spaces ...............................11

2-6. Basic concept of B-Rep model ............................................................................12

2-7. Example of typical B-Rep data structure................... ..... ....................... 13

3-1. Quadrilateral 9-node element in two different coordinate systems A) Parametric
space (r, s coordinates) B) Real space (x, y : Global coordinates)........................18

3-2. Density distribution within a 9 node quadratic element A) Density fringes B) Density
contours C ) D ensity Plot ................................................ .............................. 20

3-3. Solids defined using 3D hexahedral elements A) 8 Node element B) 18 Node
e le m e n t ........................................................................... 2 3

3-4. Editing the face represented using 2D element .............................................. .24

3-5. Three ways of editing solids represented using 3D element.................................. 25

3-6. Implicit representation of planar face using shape density function A) Density Grid
B ) C ontours of density function .................................................................... ..... 25

3-7. 3D Composition Destribution A) 8-node element B) 18-node element...................27

4-1. 2D primitive examples A) Rectangle B) Circle C) Ellipse .....................................28

4-2. Constructing a 2D primitive by the part of an existing shape .............. .................31

4-3. 2D primitives and its nodal density distribution A) Quarter circle B) Semi circle C)
Triangle D ) W edge shape............................................... .............................. 31









5-1. 2D primitive element and the corresponding extrude element...............................37

5-2. Solid created by extruding ellipse ........................................ ......................... 39

5-3. Extrusion of profile defined using multiple elements ................ .............. ..........40

5-4.. Cylindrical coordinate system for revolving .................................. ............... 40

5-5. Mapping from cylindrical to Cartesian coordinates ................................................41

5-6. Exam ples for revolved solids ............................................................................... 42

5-7. Relationship between the global and local coordinate systems...............................43

5-8. Examples of solids created using sweep elements ............................................. 45

6-1. Two neighboring straight blend elements having quadrateral 9 node elements as 2D
p ro file ............................................................................ 4 9

6-2. A straight blend element having quadratic 9 node elements as 2D profiles..............50

6-3. Cross section at a certain t* in the straight blend element........................................52

6-4. Examples of straight blend using blend elements ............................................... 55

6-5. Two neighboring smooth blend elements having quadratic 9 node elements as 2D
p ro file ............................................................................ 5 7

6-6. A smooth blend element having quadratic 9 node elements as 2D profile ...............60

6-7. Cross section at a certain t* in smooth blend elements ...........................................62

6-8. Examples of Smooth blend using blend elements.................................................67

7-1. Flow chart for set-membership classification in an individual solid element ............76

7-2 C SG tree data structure.............................................................................. ........ 77

7-3. 2D Boolean operation in CSG ............. .............................................................78

7-4. 3D Constructive Solid Geom etry Tree ............................................ ............... 79

7-5. surface normal vectors and corresponding triangles ...............................................82

7-6. Constructing a triangle from one normal vector.............................. ...............82

7-7. Constructing two triangles from two normal vectors..............................................83

7-8. Constructing three triangles from three normal vectors ............................................84









7-9. Neighborhood model using graphical method................................. ............... 86

8-1. Linear approximate step function graph............................................ .................. 88

8-2. Transform from the implicit surface function to its linear approximate step function89

8-3. Implicit surfaces defining a solid within an element in the parametric space A) 2D
solid elem ent B) 3D solid elem ent ............ .................................. ............... 91

8-4. Labeling CSG tree nodes for constructing the approximate step-function ................92

8-5. Subdivision of the 2D element and subdividing method A) Quadtree subdivision in a
2D element B) An example of quadtree representation structure C) An example of
octree representation structure ........................................... .......................... 95

8-6. A simple 2D example used for comparing the numerical integration results by
changing some parameters such as subdivision number, order of integration and
...................................... ......................................................................... 9 8

9-1. Graphics of a curve ................................ .................................... 101

9-2. Graphics of a sphere and its triangulation ..... ......... ...................................... 102

9-3. Ray methods for finding points on the boundary of the solid in 2D ........................103

9-4. Computing the points on the boundary of the ellipse...............................................104

9-5. U nion of tw o squares in 2D ........................... ................................ ............... 105

9-6. Bounding box for a 2D solid element........ ............................ ................. 106

9-7. Computing the points on the boundary of the solid A) 2D profile to be swept B)
Points on the extrude C) Points on the revolve D) Points on the sweep................107

9-8. Finding the points on the boundary of the varying cross section sweep ................ 108

9-9. 3D C SG tree structure.............................................................................. .... ........ 110

9-10. All triangleArrays composing the base and dependent solid ..............................111

9-11. Bounding cylinder method between two elements.................................................112

9-12. All triangleArrays relevant to intersected elements ..............................................112

9-13. Triangle-Triangle intersection A) Plane-Edge intersection B) Edge-Edge
intersection ............................................................... .. .... ......... 113

9-14. TriangleArrays having actually intersected triangles............................................113









9-15. Mapping the triangle in space onto the local u-v plane .............. .... ...............114

9-16. Subdivided triangles after Delaunay triangulation ...............................................115

9-17. Flow chart for constructing the graphics for the 3D Boolean result ............ ......116

A-1. SceneGraph structure for the solid modeler using implicit solid elements ............120

A-2. Class structure for the solid modeler using implicit solid elements........................122

A-3. ShapeDensity class structure ............. ............................................ .................123

A-4. InteractiveComm and class structure............................................... .................. 123

A-5. CommandList class structure.................................. ............................. 124

A-6. PositionConstraint class structure...................................................................... 124















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

SOLID MODELING USING IMPLICIT SOLID ELEMENTS

By

Jongho Lee

August 2003

Chair: Ashok V. Kumar
Major Department: Mechanical and Aerospace Engineering

A solid modeling technique using implicit surface representation is presented

where the implicit surface function (referred to here as density function) is defined by

piece-wise interpolation within quadrilateral elements in 2D and hexahedral elements in

3D. Within each element the density function is defined as a parametric function. The

solid is defined as the region within the element where the density is greater than a

threshold or "level set" value. The boundary of the solid is therefore a contour of the

density function along which the density is equal to the level set value or the boundary of

the element where the density is greater than the level set value. Simple shapes or

primitives can often be defined using a single implicit solid element while more complex

geometry can be represented either using a grid/mesh of elements or by Boolean

combination of simple primitives. The geometry of the primitives can be edited or

modified by changing the density values at the nodes of the elements, by changing the

size of the element as well as by changing the level set value. In this study, hexahedral









elements are presented along with basis functions that can be used to represent primitive

solids created by sweeping 2D geometry along trajectories that can be lines, arcs or

arbitrary parametric curves. In traditional solid modeling software based on the B-Rep

(boundary representation) approach, solid primitives are often created by sweep

operations. The elements described in this study enable the representation of such solids

using implicit surfaces. This primitive representation scheme is axes independent because

the implicit function representing the solid is defined as a parametric function within

quadrilateral or hexahedral elements while general implicit surface functions are axes

dependent. In general it is easier to determine whether a given point is inside, outside or

on the boundary of a solid if the boundary is represented using implicit equations. Since it

is necessary to repeatedly classify points in this manner for many graphical display and

volumetric property evaluation algorithms implicit representation is particularly suited

for representing primitives in a CSG tree. For the implicit solid element representation, it

is also necessary to first map from global coordinates to parametric coordinates to

perform this classification. This mapping is shown to be easy to perform if the elements

have parallel edges and faces. The volume of the solid can be computed by defining the

linear approximate step function, which has a unit value almost everywhere in the interior

of the solid and zero in exterior of the solid. This linear approximate step function can be

numerically integrated to compute the volume.














CHAPTER 1
INTRODUCTION

1.1 Overview

Solid modeling is the fundamental technology on which CAD/CAM/CAE

(Computer Aided Design/Manufacturing/Engineering) systems have been built. Solid

modeling technique (sometimes called volumetric modeling) has been used for many

applications including visualization, geometry design, assembly verification, engineering

analysis and generating NC machining codes. The use of solid models for design and

manufacturing is becoming more widespread with the increasing availability of computer

technology.

Solid models enable one to construct unambiguous models of three-dimensional

objects. Such unambiguous models are necessary to construct algorithms for

automatically computing volumetric properties such as surface area and volume of the

solid. Algorithms for automatic generation finite element mesh for engineering analysis

also depend on the ability to construct precise, unambiguous models of solids.

Algorithms have also been developed to generate tool paths for machining given the solid

model of the part to be machined. More recently developed fabrication and prototyping

technology called the layered-manufacturing technique (also known as rapid prototyping

or the solid freeform fabrication technique) also need solid models of the part to be

constructed.

The most popular schemes in current solid modeling are Boundary Representation

(or B-Rep) and Constructive Solid Geometry (CSG). In B-Rep, a solid is represented by









the boundary information of the solid where vertices, edges and faces (geometry

information) are saved together with the information on how they are connected to build

the boundary of the solid (topology information). The geometry of the boundaries are

represented using parametric equations of the form X(u) or X(u,v) where X is a position

vector of a point on the curve or surface. In CSG representation, Boolean operations

applied on primitives that are simple shapes such as a block, cylinder, sphere or cone are

saved in the binary tree data structure where primitives are defined by Boolean

combination of half-spaces. Half spaces are implicit equations of curves or surfaces that

are represented as f(x,y) > 0 (curves) or f(x,y,z) > 0 (surfaces).

Current commercial solid modeling systems typically use a hybrid system where

the solid is constructed by the Boolean combination of primitives that are represented

using B-Rep models. The B-Rep model of the resultant solid is automatically constructed

from the CSG tree using algorithms known as Boundary evaluation.

In almost all current solid modeling systems, a feature based approach is provided

as the user interface for creating solids where primitives are constructed by sweeping 2D

profiles along a trajectory. Such sweep features for creating 3D primitives are popular

because it enables the user to create primitives interactively. B-Rep solid models use

parametric equations for geometric components such as lines, curves and surfaces.

Creating B-Rep primitives by sweep operations is simple and straight forward because it

is easy to define the equations of surfaces created by sweeping a curve along a trajectory

in the parametric form. However, it is difficult to construct such sweep primitives by a

Boolean combination of half-spaces as is done in traditional CSG systems.









Solid modeling systems based on B-Rep models is very difficult and expensive to

implement. The main reason for the difficulty is that the B-Rep model for the resultant

geometry has to be constructed automatically from B-Rep models of the primitives using

the information about the Boolean operations used to combine them. For each regularized

Boolean operation, the boundary evaluator algorithm, detects intersections between

participating solids, computes intersection geometries (intersection points and curves as

well as sub-divided faces) and classifies them according to whether the geometry is in, on

or outside the final solid defined by the Boolean operation. As mentioned earlier, B-Rep

models need topology information as well as geometric information. Constructing

connectivity (topology) between geometric components automatically is very difficult to

be performed for complex geometries. And it is also very expensive to classify a given

point, curve or face as being inside, outside or on the solid. The procedure for such

classification is called set-membership classification.

In traditional CSG models, the topology of the solid is not explicitly represented.

Therefore, there is no need for Boundary evaluation algorithms. An algorithm is needed

to generate the graphics of the resultant solid using the information available in the CSG

tree. Even though in general the graphics image is difficult to generate for implicit curves

and surfaces, the implementation of a CSG solid modeling system is easier because there

is no need for Boundary evaluation algorithms. However, in order for such a modeling

system to be useful it is necessary that a feature based modeling interface can be

constructed. No attempt has been made to construct sweep features using implicit curves

and surfaces since all current solid modeling systems use B-Rep models. Therefore one









of the motivations of this study is to develop a feature based solid modeling system

where all the boundaries of the solid are represented using implicit equations.

1.2 Goal and Objectives

The goal of this research is to study the feasibility, advantages, and disadvantages

of building a feature based solid modeling system that uses only implicit equations for

curves and surfaces as it is done in traditional CSG solid modeling systems. Such solid

models in this thesis are referred as implicit solid models. In order to develop such a

system, implicit solid elements that are quadrilateral and hexahedral elements used to

represent implicit curves and surfaces respectively have been used. The main objectives

of the thesis are listed below.

1. Develop quadrilateral implicit solid elements to represents simple 2D primitives such
as circles, ellipses, rectangles and triangles.
2. Develop hexahedral implicit solid elements to represent solid primitives obtained by
extruding, revolving or sweeping 2D primitives defined using quadrilateral implicit
solid elements.
3. Develop hexahedral implicit solid elements to represent a solid primitive obtained by
blending between two or more different 2D primitives
4. Construct algorithms for displaying solids created by Boolean combinations of
primitives represented using implicit solid elements.
5. Construct algorithms for automatically computing volumetric properties.
1.3 Outline

In Chapter 2, current popular solid modeling techniques are introduced to give an

overview for the reader. In Chapter 3, the definition of implicit solid elements is

described and in Chapter 4, simple 2D primitives using 2D implicit solid elements are

introduced. In Chapters 5 and 6, constant cross-section sweep elements and variable

cross-section sweep elements are described respectively. Algorithms for set membership

classification at a given point with respect to a primitive and a solid represented by a

CSG tree structures are described in Chapter 7. In Chapter 8, the method to compute the






5


volume of the solid is presented and in Chapter 9, issues associated with generating the

computer graphics of the solid are described. Finally, Chapter 10 provides conclusions

and discussions.














CHAPTER 2
SOLID REPRESENTATION SCHEMES

2.1 Geometric Modeling

Geometric Modeling systems are used in the design process for visualization,

assembly verification, engineering analysis, generating NC machining codes, etc. It often

eliminates the need for making many physical prototypes to test design concepts. There

are three different types of geometric modeling systems: wireframe modeling, surface

modeling and solid modeling.

Wireframe Modeling is the earliest method, where a shape is represented by its

characteristic lines and points. Although it has some advantages such as simple user

inputs and ease of implementation, the following two disadvantages make this

representation unpopular. First, the shapes represented only by lines and points are very

ambiguous. Second, there is no information about the inside and outside boundary

surfaces of the object. Therefore it cannot be used for mass property computation, tool

path generation or finite element analysis.

Surface Modeling has surface information in addition to the wireframe model

information. Therefore, the tool path for the NC machines can be generated for the

surfaces. However, since it also does not have information to distinguish between the

inside and the outside of the object, Boolean operations cannot be used to combine such

models and it is difficult or impossible to construct other algorithms for automatically

manipulating such models as an example when generating a mesh.









SolidModeling was introduced to overcome some of the limitations of the other

schemes mentioned above. Solid modeling techniques began to develop in the late 1960s

and early 1970s. The fundamental concepts and definitions of solid modeling are well

introduced in many books [Hoffmann, 1987, Zeid, 1991., Mortenson, 1997., Lee, 1999].

The most popular schemes in current solid modeling are Boundary Representation (or B-

Rep) and Constructive Solid Geometry (CSG). In B-Rep, a solid is represented by the

boundary information of the solid where vertices, edges, and faces are saved together

with the information on how they are connected to build the boundary of the solid. In

CSG representation, Boolean operations applied on primitives are saved in the binary tree

data structure. These two representation schemes are described in the following section.

2.2 Constructive Solid Geometry (CSG)

The constructive solid geometry representation technique defines complex solids as

Boolean combinations of simpler solids called primitives. A Boolean operation is one of

the best methods to create complex solids. It dramatically increases the repertoires of the

shapes to be modeled. A binary tree data structure is used to save the Boolean operations

between primitives. The following is an example of CSG tree data structure, where the

final solid is generated by the union of the primitive A and B, followed by the difference

of the primitive C.

In this tree structure, leaf nodes (or primitive nodes) represent primitive solids and

branch nodes have Boolean types such as union, difference, and intersect. Boolean

operations can be explained by set theory. However, the set theory applied here is

different from the original set theory. A modified Boolean operation has been defined for

solid modeling which is referred to as Regularized Boolean operation, where the Boolean

combination results between primitives are closed and dimensionally homogeneous.


















A B





Figure 2-1. Example of CSG binary tree structure

The following example shows how the regularized Boolean operation is different

from the original Boolean operation. Regularized Boolean operations are denoted by u ,

n*, and -* respectively while the ordinary Boolean operations are denoted by u, n, and






AA



Union Intersection Difference
*





Figure 2-2. Comparison between the ordinary Boolean and regularized Boolean operation

As shown in Figure 2-2, problems always occur when the boundaries of the

primitives overlap. These special cases can be treated to create a regularized result by

comparing the directions of both boundary normal vectors. For example, in the

regularized intersection, if both boundary normal vectors have the same direction, the









boundary is also the boundary of the regularized Boolean combination. On the contrary,

if they have opposite directions, it should be removed to construct the regularized

Boolean combination. The following shows all-special cases, where nA and nB are the

boundary normal vectors of the primitive A and B at the overlapped part respectively.


u ON IN
nON OUT
-OUT ON
Figure 2-3. Regularized Boolean treatment in overlapped case

The method using normal vectors for the regularized Boolean combination in solid

modeling cannot be used when multiple normal vectors are involved. For instance

vertices in 2D or 3D and edges in 3D cannot be simply classified as in, out or on the solid

because those have multiple normal vectors. This problem can be solved by using a

neighborhood model where points close to the given point (or in its neighborhood) are

examined. Figure 2-4 shows an example of a neighborhood model where a small disk

symbolizes the neighborhood, and the shaded area indicates points inside the solid. The

neighborhood model at a point for the Boolean result is constructed by applying the

Boolean operation to the neighborhood models at the point for the two participating

solids.










On
AUB



A ( A B
B
On
AUB



A B
Figure 2-4. Examples of neighborhood models on the vertices in 2D

There are three important subsets defined by a solid, where the solid is considered a

set of points. Those are the set of interior points, points on the boundary of the solid, and

all points outside the solid. Set-membership classification of a point involves assigning

the point to one of these sets. It is an essential step to do a Boolean operation.

Primitives are simple shapes such as a block, cylinder, sphere or cone, where the

mathematical descriptions of the primitives are defined by the intersection of a set of

curved or planar half-spaces. A half space divides the space into two regions, for

instance, an infinite x-y plane divides the space into the region of z>0 and the region of

z<0 as shown in Figure 2-5A. Figure 2-5B and Figure 2-5C show how to define the

primitive block and cylinder by half-spaces respectively. Where the block is represented

by the intersection of six planar half-spaces and the cylinder is represented by the

intersection of a cylindrical half-space and two planar half-spaces.














z b

xxa

Sy>b


(A) (B) (C)


Figure 2-5. Mathematical description of the primitives by half-spaces A) Half-space by x-
y plane B) Block by half spaces C) Cylinder by half spaces

Arrows indicate the direction of the material. The detailed mathematical

descriptions can be represented as follows:

Block: a
Cylinder : 0 < x2 + y2 r2, a
This kind of description is good for set membership classification because the sign

of the implicit function representing the half space can be used to classify the point. CSG

representation does not include topology information and therefore additional

computation is necessary to determine the connectivity between boundary entities. Such

connectivity information is used in many applications such as mesh generation and tool

path generation for machining.

2.3 Boundary Representation (B-Rep)

The B-Rep model of a solid consists of parametric equations of the vertices, edges

and faces of the solid together with the connectivity information between them. The

geometry information in a B-Rep model is composed of surface equations, curve









equations, and point coordinates. The connectivity between the geometric entities is the

topology information which is the interrelationship among faces, edges and vertices.

It is based on the idea that the boundary of a solid consists of faces (surfaces)

bounded by edges (curves), which in turn are bounded, by vertices (points). In B-Rep, a

face is a subset or limited region of some more extensive surface. For example, planar

faces are subsets of infinite planes that are bounded by edges, while curved faces are

represented as parametric surfaces bounded by edges. Thus, these equations are saved in

the B-Rep data structure. The topology information is represented as relations between

faces, edges, and vertices. In Figure 2-6, the face Fi is composed of four edges (El, E2,

E4, E3) and each edge is represented by two vertices.
El
V2 V1

F2 Fl E2 Fl E4

V3 V4
E3

Figure 2-6. Basic concept of B-Rep model

An array of faces, edges, and vertices can be used to represent this kind of simple

polyhedral model. Figure 2-7 shows the typical data structure [Spatial Technology Inc.,

1995] used to represent B-Rep models which consists of shells, faces, loops, half

edges(or coedges), and vertexes.

Figure 2-7 shows classes used to represent the solid model. A solid is denoted as a

BODY which consists of one or more LUMPs. A LUMP is a connected 3D region whose

boundaries are represented using closed SHELLs. A closed SHELL consist of a list of

faces forming a closed volume. As mentioned earlier, the geometry of a face is defined

by a parametric surface equation and is bounded by LOOPs that define its external and

internal boundaries. Loops are defined as a list of half-edges (COEDGEs), which are










edges with a direction. Half edges are connected together to form loops such that the

direction of the loop is counter clockwise for the external boundary and clockwise for

internal boundaries when the face is viewed from the outside. Assigning directions to

loops in this manner assists in determining whether a given point is inside, on, or outside

the solid. If the solid has a lot of faces, this kind of set membership classification is

difficult and it requires significant computation. Since every entity is interconnected with

other entities, changing the topology is very complicated in the B-Rep model approach.


Ex)
BODY Ex)T
Fl
LUMP Lumpl- S1


SHELL S1- F1, F2 ....... F10


SURFACE FACE next F1 L1, L2 L1
"L2
LOOP next L11C1, C2, C3, C4

next V2
COEDGE previous C1E1 -E

El
CURVE EDGE E1V1, V2


POINT VERTEX


Figure 2-7. Example of typical B-Rep data structure

2.4 Hybrid System

Creating B-Rep models directly by defining each vertex, edge, and surface is

extremely cumbersome. Therefore, hybrid systems were developed where the solid is

represented procedurally using a CSG tree but the primitives are represented using B-

Rep. The B-Rep model of the solid represented by the CSG tree is evaluated









automatically using "Boundary Evaluator" algorithms [Requicha and Voelcker, 1985].

For each regularized Boolean operation, the boundary evaluator algorithm, detects

intersections between participating solids, computes intersection geometries (intersection

points and curves as well as sub-divided faces) and classifies them using set membership

classification algorithms to determine whether the geometry is in, on, or outside the final

solid defined by the Boolean operation. While set membership classification is easy when

the solid is represented as a combination of half spaces (where the surfaces are implicit

equations, f(x,y,z) = 0), it is more difficult to do for B-Reps. Furthermore, constructing

the topology of the resultant solid automatically is also difficult. As a result it is very

expensive to build robust and reliable software that can handle every special case.

However, over the years many commercial systems have been developed that have very

reliable boundary evaluator algorithms that are robust and can handle almost all special

and degenerate cases including non-manifold topologies. To cope with the high cost of

implementing these algorithms, many companies in the CAD industry buy "geometric

modeling kernels" from other companies that have already implemented them [Spatial

Technology Inc., 1995].

2.5 Sweep Features

A sweep feature is defined by sweeping a planar shape along an arbitrary space

curve referred to as a sweep trajectory, where the cross section of the sweep solid can be

either constant or varying. This is a very popular and useful method due to the fact that

the solid represented by sweeping is simple to understand and execute. If the sweep

trajectory is a straight line, it is called as translational sweep (or extruded solid). When

the sweep trajectory is a circular arc, it is termed a rotational sweep (or revolved solid).

When the sweep trajectory is an arbitrary parametric curve, it is referred to as a general









sweep. Recent research related to sweeping includes three-dimensional object sweeping

and sweep surface representation using coordinate transforms and blending [Martin and

Stephenson, 1990, Choi and Lee, 1990].

2.6 Implicit Surfaces or Level Set Surfaces

Traditionally, implicit curves and surfaces are represented as f(x) = c where x e R2

for planar curves and x e R3 for surfaces. They are sometimes referred to as level sets or

iso-curve/surface since the curve or surface corresponds to a constant value of the

function f(x). If c = 0 then the curve or surface is called the zero set of f(x). An implicit

surface divides space into regions where f(x) > 0 and f(x) < 0. Therefore, if the

boundaries of the solid are represented implicitly as half-spaces one can use the sign of

f(x) to perform set membership classification that is to determine whether a given point is

inside, outside, or on the solid. Despite this advantage, implicit representation for curves

and surfaces has traditionally been considered inferior to parametric representation. The

reasons suggested for this include axis dependence of implicit curves, difficulty in tracing

or generating graphics, as well as difficulty in fitting and manipulating freeform shapes.

Considerable progress has been made to solve some of these problems due to which

implicit curves and surfaces now have numerous applications in graphics and animation

[Bloomenthal et al., 1997].

To enable graphical display a variety of polygonization, tessellation, or tracing

algorithms have been developed [Wyvill and Overveld, 1997., Lorensen and Cline,

1987]. Ray tracing algorithms have also been used for visualization of implicit surfaces

[Glassener, 1989]. When the implicit function f(x) is a polynomial, the surface is called

an algebraic surface. Most common primitive shapes such as sphere, ellipse, cone, and









cylinders can be expressed as algebraic surfaces using quadratic polynomials. For more

complex shapes, implicit algebraic surface patches (or A-splines) have been developed

[Bajaj et al., 1995]. These patches can be used for C1 and C2 interpolations or

approximations and also for interactive free-form modeling schemes. Implicit curves and

surfaces have also been used for visualization problems such as shape reconstruction

from unorganized data sets [Zhao et al., 2000] and dynamic fluid flow [Osher and

Redkiw, 2003]. Recently, R-functions [Shapiro, 1998] have been used to define implicit

solids. The sign of an R-function depends only on the sign of its arguments and not its

magnitude. This property of R-functions can be used to construct implicit functions for

representing solids created by a Boolean combination of implicit surfaces. Furthermore,

the advantage using geometry represented using implicit curves and surfaces in solving

boundary and initial value problems with time varying geometries and boundary

conditions has also been illustrated [Shapiro and Tsukanov, 1999].

Implicit surfaces are often used in computer graphics for representing soft or

deformable objects such as humans, animals, and amorphous blobs. Sometimes, an

implicit surface can be represented by the distance relationship between the surface and a

given basic structure such as point, line, and circle. This kind of method is called distance

metrics or the skeleton method [Tigges and Wyvill, 1999].
















CHAPTER 3
IMPLICIT SOLID ELEMENTS

3.1 Definition of Implicit Solid Elements

The implicit curves and surfaces are represented here as the level set of an implicit

surface function 4(r,s) = kb for planar faces or 4(r,s,t) = kb for solids. The implicit

surface function, 4, has been referred to in this study as the shape density function (or

simply density function) [Kumar and Gossard, 1996]. The density function has a value

greater than a threshold or level set value, )b, inside the solid and less than kb outside the

solid. Unlike the traditional approach, where the implicit surface function is defined

directly in terms of the x, y, z or Cartesian coordinates, the density function used here is a

parametric function whose arguments are the parametric coordinates of an implicit solid

element. This is illustrated using a two dimensional example in Figure 3-1 where the

element has nine nodes and its geometry in the parametric space is shown in Figure 3-1A

while its real geometry is shown in Figure 3-1B. In the parametric space the element is a

square whereas in the real space the element can be a distorted quadrilateral. The

mapping between the parametric coordinates and the real coordinates x, y and z is defined

as,


n
x(r,s,t) = x, M,(r,s, t),
1=1
n
y(rs,t)= y1M1(r,s,t) and
1=1
n
z(r,s, t) = zM (r,s, t)
1=1 (3-1)










In Eq. 3-1. (x,, y,, z,) are the coordinates of the node i, M,(r,s,t) are mapping basis

functions that define the mapping between the parametric space (r,s,t) and the real space

(x,y,z) and n is the total number of nodes per element.


s 51
2 5 2
21 = b 8

8 r
Y 4
3 7 4 x3


(A) (B)

Figure 3-1. Quadrilateral 9-node element in two different coordinate systems A)
Parametric space (r, s coordinates) B) Real space (x, y : Global coordinates)

The density functions are defined within the elements by specifying the values of

the functions at the nodes of these elements. The value of the function within each

element is obtained by interpolating the values at the nodes using appropriate basis

functions or interpolation functions. A parametric interpolation scheme is used as shown

below where parameters (r, s, t) vary between -1 and 1.

n
q(r,s) = N, (r, s) for 2D elements (3-2)
1=1
n
q((r,s,t)= 0q,N, (r,s,t) for 3D elements (3-3)

Where, I, is the density at the node i, Ni is the interpolation basis function at the

node i used to interpolate the density, and n is the total number of nodes in the element.

Each element is a cube (or square in 2D) of side length equal to 2 in the parametric space

since the parameters r, s and t vary from -1 to 1 within each element. If the mapping basis

functions Mi (r, s, t) are identical to the density interpolation basis functions Ni(r,s,t)









then the element can be referred to as "iso-parametric element". However, it is often

beneficial to use simpler basis functions for the mapping.

In this formulation, even though the boundary of the solid is represented in the

implicit form 4(x,y,z) = 4b, the density function itself is represented in a parametric form,

4(r, s, t) with a mapping defined between the parameters (r,s,t) and the global coordinates

(x,y,z). Therefore, many of the advantages traditionally associated with parametric curves

and surfaces are also applicable to the implicit solid element approach including axes

independence and the ease in tracing. The parametric form of the density function enables

the development of simple algorithms to polygonize the surface for graphical display.

The detailed description for one way of graphical display used in this system is explained

in Chapter 9.

The following table shows the basis functions for the 9 node quadrilateral element

where the numbering system is as shown in Figure 3-1.

Table 3-1 Basis functions for the 9 node quadrilateral element
N1 = 0.25(r2+r)(s2+s) N2 = 0.25(r2-r)(s2+s) N3 = 0.25(r2-r)(s2-s)
N4 = 0.25(r2+r)(2-s) N5 = 0.5(1-r2)(2+s) N6 = 0.5(r2-r)(1-s2)
N7 = 0.5(1-r2)(2-s) N8 =0.5(r2+r)(-s2) N9 = (1-r2)(1-s2)


These shape functions are derived by Lagrange interpolation such that Ni is, equal

to 1 at node i and is zero at other nodes. For instance, N1 = 0 at node 2 to 9 is equivalent

to requiring that N1 = 0 along edges r = 0, r = -1, s = 0 and s = -1. Therefore N1 can be

represented by

N = c(r2+r)(s2+s)
where c is some constant that is determined from the condition N1 = 1 at node 1.

Since r = 1 and s =1 at node 1, yields









1 = c(12+1)(12+1)= 4c
Which yields c = 0.25.

Similarly other shape functions at nodes 2 to 9 are derived as shown in Table 3-1.

Figure 3-2 shows the density distribution within a 9-node element when the density

at the corer nodes are set equal to 0, the density at mid-edge nodes are set to 0.5 and the

density is 1 at the center node.





:/




(A) (B) (C)

Figure 3-2 Density distribution within a 9 node quadratic element A) Density fringes B)
Density contours C) Density Plot

In Figure 3-2B each contour line corresponds to constant shape density values.

Therefore the boundary of the solid in this system is one of the contours whose value is

equal to the boundary density value (b ). For instance if 0.5 is set to Ob, the boundary of

the solid would be a circular shape as highlighted in Figure 3-2A.

3.2 Definition of the Solid within an Implicit Solid Element

In brief, the definition of the solid used here can be represented as # > Ob where q

is a density value defined at the isoparametric coordinates namely, -1 < r < 1 and

-1 < s < 1 in 2D (and -1 < t < 1 in 3D). Therefore, the solid in this system may now be

defined as the set of points S such that











S= X _= ,NJ(r,s)>b, X X,M,(r,s) for2Delements
S for 2D elements
where -

S= X= ,N r's, t) b X= XM (r,s,t) for 3D elements (3-4)
1-i 1=1
where -1 where, X e R2 for 2D elements and X e R3 for 3D elements.

Since the boundary of the geometry is very critical in some cases, clear definition

of the boundary of the solid is required in any solid modeling system. In this system there

are two criterions defining the boundary of the solid. Therefore, the boundary of the solid

represented by an element can be defined as the set of points B such that



X~ =ZN,(r,s)=-b, X= XM,(r,s)
1=1 1=1
where -l B = AND for 2D elements

1=1 1=1
where r = +1 ors = +1

n n
= SN,(r,s,t) =4b, X = XM, (r,s,t)
1 1 1=1
where - B= AND for 3D elements (3-5)

X. = 0N, (r, s, t) >b, X= X,M, (r, s, t)
z=1 1z=
where r= lors =l ort =+

3.3 3D Hexahedral Solid Elements

Three-dimensional primitives can be defined using hexahedral elements such as

hexahedral 8-node or 18-node elements. Figure 3-3 shows these two 3D elements and the

node numbering scheme used. The basis functions of a hexahedral 8-node element and

the interpolation within each element are trilinear. Therefore, the higher order 18-node









element is required to create cylindrical and conical shapes. The basis functions of the 8-

node and 18-node hexahedral element are listed in the following tables.

Table 3-2 Hexahedral 8-node basis functions
N1 = 0.125(1+r)(1+s)(1+t) N2 = 0.125(1-r)(1+s)(l+t) N3 = 0.125(1-r)(1-s)(l+t)
N4 = 0.125(l+r)(1-s)(l+t) N5 = 0.125(1+r)(1+s)(1-t) N6 = 0.125(1-r)(1+s)(1-t)
N7 = 0.125(1-r)(1-s)(1-t) N8 = 0.125(1+r)(1-s)(1-t)

Table 3-3 Hexahedral 18-node basis functions
N1= 0.125(l+r)(l+s)(l+t)rs N2= -0.125(1-r)(l+s)(l+t)rs N3= 0.125(1-r)(1-s)(l+t)rs
N4= -0.125(1+r)(1-s)(l+t)rs N5= 0.125(1+r)(l+s)(1-t)rs N6= -0.125(1-r)(l+s)(1-t)rs
N7= 0.125(1-r)(1-s)(1-t)rs N8= -0.125(1+r)(1-s)(1-t)rs N9= 0.25(1-r2)(1+s)(1+t)s
N10= -0.25(1-r)(1-s2)(l+t)r N1= -0.25(1-r2)(1-s)(1+t)s N12= 0.25(1+r)(1-s2)(l+t)r
N13= 0.5(1-r2)(1-s2)(l+t) N14= 0.25(1-r2)(1+s)(1-t)s N15= -0.25(1-r)(1-s2)(l-t)r
N16= -0.25(1-r2)(1-s)(1-t)s N17= 0.25(1+r)(1-s2)(1-t) N18= 0.5(1-r2)(1-s2)(l-t)


Figure 3-3 shows two solids defined using these hexahedral elements with the

boundary density value 4b = 0.5. In Figure 3-3A, an 8-node element is used with the

following values of the nodal densities: 1=0.4 Q 2 =1.0 4 3 =0.2 4 4 = 1.0 5= 0.4 46 =

0.9 0 7 = 0.3 8s = 1.0. Figure 3-3B shows a cylindrical solid with the elliptical cross-

section created using the 18-node element. The density values at the nodes were set as

follows to create this geometry: 1= 2= 3 =4 =5 =6= 7 = 8=0.0, 9 =010=o 11

= 12 = 14 =15= 16 =17=0.5, and 13 =18 1.





3 t 10
3 t 11 13 2
2 1 s
412
o I 86 -- 1-- 6
6 18 14
0 r 8 17 5




















(A) (B)

Figure 3-3 Solids defined using 3D hexahedral elements A) 8 Node element B) 18 Node
element

In the following section, the method to define and edit primitives using implicit

solid elements is described.

3.4 Defining and Editing Primitives

The geometry defined within each element can be edited by modifying the density

values at each node, the shape of the element. or level set value of density. This diversity

in editing methods enables the creation of a variety of primitives using a single element.

Figure 3-4 shows how the solid geometry is changed in response to the change in such

parameters as density value at each node, nodal coordinates of the element, and boundary

density value. In Figure 3-4A, the density value at the nodes of the 9-node element is set

such that the density at all nodes is set to 4b = 0.5, except the central node where density

is set equal to 1. The geometry is therefore identical to that of the element since every

point within has density greater than 4b. The geometry in Fig. 3-4B is obtained by

changing the density values at nodes to 1 =0.2, 0 2=0.3, 0 3 =0.4, 0 4 =5= 6 = 7 =08

= 0.5 and 9=1.

Changing the shape of the element as shown in Figure 3-4C where the element

nodes are moved to stretch the element into a rectangle can also modify the geometry.








Note that the element shape should be a parallelogram in this system. This method of

editing the geometry enables a variety of quadrilateral primitives, such as rectangles,

trapezoids and parallelograms, to be constructed by deforming the element in

Figure 3-4A.





Em--

(A) (B) (C) (D)
Figure 3-4 Editing the face represented using 2D element

Yet another way to modify the geometry is by changing the level set value, 4b, so

that a different level set or contour of the density function becomes the boundary of the

solid. This is illustrated in Figure 3-4, where the solid in Figure 3-4D is obtained from the

solid in Figure 3-4C by changing the level set value from 0.5 to 0.6.

Figure 3-5 shows the above three methods of editing for a 3D element. The solid in

Figure 3-5B is obtained by just changing the density values of the solid in Figure 3-5A

and it is further modified by changing the shape of the element to obtain the solid in

Figure 3-5C. Finally, the solid in Figure 3-5D is obtained by modifying the level set

value 0b.



















(A) (B) (C) (D)
Figure 3-5 Three ways of editing solids represented using 3D element

3.5 Defining Complex Geometries using Implicit Solid Elements

Simple solid primitives may be defined using a single implicit solid element. Using

a grid of elements as illustrated in Figure 3-6, it is possible to create more complex

primitives. Figure 3-6A shows the two-dimensional mesh or grid used to represent the

shape and Figure 3-6B shows the density values using gray scale such that white is ) = 1

and black is 4 = 0. The boundary of the geometry is represented partly by the contours of

the density function ) = 0.5 (which is highlighted) and partly by the boundaries of the

mesh where ) > 0.5. When the primitive is modeled using a mesh that has many elements

as in Figure 3-6, the solid model is the union of the solids defined by the individual

elements in the mesh.











(A) (B)
Figure 3-6 Implicit representation of planar face using shape density function A) Density
Grid B) Contours of density function









A variety of elements and basis functions can be defined and used to represent the

solids in 2D and 3D. Higher degree polynomials can be used as basis functions for

elements with a larger number of nodes per element. If the geometry to be represented is

simple (e.g. rectangle, circle or ellipse) then it is often possible to define the required

shape density function using just one element. Primitives thus defined using one or more

elements can be combined using various Boolean operations to create a procedural

definition of more complex solids, expressed as a CSG tree. Refer to Chapter 7 for the

detailed description about CSG representation scheme used in this system.

3.6 Heterogeneous Solid Model Capability

Conventional solid modeling systems have been developed based on the

assumption that the solid is perfectly homogeneous. Thus they cannot represent internal

properties or offer ways of representing internal behavior. In this solid representation

using implicit solid elements, material composition can be defined within each element

by interpolating nodal values of composition. Many biomechanical structures such as

bones, shells, and muscular tissues are heterogeneous. Once heterogeneous objects are

modeled as the solid models, they can be manufactured by recently developed

manufacturing methods such as layered manufacturing technique, also known as rapid

prototyping or solid freeform fabrication. The following examples illustrate this idea

where material composition distributions are represented within solids by defining

composition values at each node and using the same interpolation scheme used in

defining geometries.



















(A) (B)
Figure 3-7 3D Composition Destribution A) 8-node element B) 18-node element

In this case, two materials termed A, B respectively are used for composing the

heterogeneous objects. The A material is shown as red in Figure 3-7 and the material B is

shown as blue. Let be the volume percentage of material A at a point. Then 1- is the

volume percentage of B at the same point. The nodal density values are larger than the

boundary density value (0 b) for both examples in Figure 3-7. The material compositions

(volume percentage of material A) at the nodes for creating Figure 3-7 were defined as

follows:


(A) 1= 2= 5= 6=1.0 53= 4= 7 =8=0.0
(B) 1=52=3 = 4 =5 =6= 7 =8=0.0,
S9 =10= 11= 12= 14= 15= 16= 17=0.5, and < 13= 18= 1 (3-6)















CHAPTER 4
TWO DIMENSIONAL PRIMITIVES

As mentioned in the proceeding chapter, simple primitives are combined using

Boolean operations to create more complex solids. In this chapter, 2D implicit elements

used for constructing 2D primitives are presented.

4.1 2D Primitives by 9-Node Quadrilateral Elements

A quadrilateral primitive can be represented directly using quadrilateral elements

by setting the nodal values of density to be greater than or equal to the level set value.

While it is possible to create a variety of implicit solid elements, the 9-node quadrilateral

element for two-dimensional primitives such as quadrilaterals, circles and ellipses has

been used primarily. A variety of elements and interpolation functions can be used for

defining primitives however for brevity and consistency a 9-node element was chosen

among lots of capable elements to create all primitives in this system.

The following figures show examples of 2D primitives defined by the quadratic 9-

node element.



N '







(A) (B) (C)

Figure 4-1 2D primitive examples A) Rectangle B) Circle C) Ellipse









Rectangular primitives as shown in Figure 4-1A can be defined by giving density

values larger than the boundary density value (4b) at each node. Then any rectangular

shapes can be created by modifying the size of the element.

A primitive circle can be defined using a single 9-node quadrilateral element. The

boundary is defined as

9
4(r,s)= N, (r,s)b, = =0.5 (4-1)

Expanding the above equation by substituting N, listed in Table 3-1, yields

0.5= N,11 +N2 2+N3 3+N4 4+N55 +N6 6+N707+ N88+ N9 9 (4-2)
= 9 + (0.56 -0.5 8)r + (-0.5 5+0.5 7)s+ (0.5 6 +0.5 8- 9)r2
+ (0.5 5 +0.5 7 9)s2 + (0.25 3 -0.25 2 -0.25 4 -0.25 41 )rs
+ (-0.25 2+0.55 +0.2504+0.2503 0.25 1-0.5 7 )r2 s
+ (-0.25 1+0.25 2 +0.25 3-0.25 4 0.5 6 +0.5 8s )r s2
+ ( 9 +0.25 1 -0.5 5 -0.5 6 + 0.25 4+0.25 3+0.25 2-0.5 7 -0.5 8)r2 s2
To be an equation of circle, the coefficients of r, s, rs, r2s, rs2 and r2s2 should be

equal to zero and the coefficients of r2 and s2 must be same. In other words, the following

conditions should be satisfied.

(0.506 -0.58) = 0
(-0.5 5+0.57) = 0
(0.25 3 -0.25 02 -0.25 4 -0.25 1 ) = 0
(-0.25 2+0.55 +0.2504+0.2503 0.25 1-0.5 7)= 0
(-0.25 1+0.25 2 +0.25 3-0.25 4 0.5 6 +0.5 s ) = 0
( 9 +0.25 1 -0.5 5 -0.5 6 +0.25 4 +0.25 3+0.25 2-0.5 7 -0.5 8) = 0
(0.506+0.508-09) = (0.505+0.507-09) (4-3)
From the above equations, the following conditions are derived.

1= 42 =3 =4
45= 6 =7 =08









49=205 -01
4 9 45 (4-4)
This results in a circle of unit radius within the element in the parametric space

whose equation is r2 + s2 = 1. The geometry in the real space will also be a circle if the

shape of the element is square and diameter of the circle will be equal to the size of the

square as shown in Figure 4-1B. If the element shape is a rectangle in real space, then the

circle in the parametric space maps into an ellipse in the real space with the major and

minor diameters equal to the length and height of the rectangle as shown in Figure 4-1C.

Clearly there is no unique representation for any given primitive. In the following

examples quadrilateral 9node elements are used for all primitives.

Table 4-1: Samples of 2D primitive design
4b Element 4 i (Nodal densities)
shape
Rectangle 0.5 Square 1= 2 = 3 = 4 = 5= 6 = 7 = 8 = 0.5 and 9=1
Circle 0.5 Square ( 1= 2= 3 =4 =0.0, 45=0 6 =07 =8 = 0.5 and 4 9=1
Ellipse 0.5 Rectangle 1 =02= 3 =4=0.0, 05= 06 =07 =8 = 0.5 and 49=1


Figure 4-2 shows more 2D primitive examples created using the 9-node elements.

The value of density at the node is displayed near each node. The density values at each

node can be found using the same approach that was used for computing the nodal

density for the circle. Alternatively, one can make use of the property of Lagrange

interpolation that the interpolation obtained is unique. Therefore if the implicit equation

of a curve is known then the value of the function evaluated at the nodal coordinates can

be assigned as the nodal values.












2 = 0 0= 0.5 = 0





=0.5" = 0.5 6 =
S=0 0= 1




3 =o0 =0.5 #4 =o


2 = #(0,1) = #(0.5,1) A = (1,1)





:0,0.5) = #
= (0.5,0.5)


0) A = (0.5,0) 04 = (1,0)


Figure 4-2 Constructing a 2D primitive by the part of an existing shape


Figure 4-2 shows how to compute nodal density values for a quarter circle


primitive from the primitive circle where the density function used is as follows:

9 1
(r,s)= N,(r,s)O =1--r --s
11 2 2


06 -0


=2 0.5 ,= 0.375




.875,, =
=9 0.75



S=1 7= 0.875 4 =0.5


^ = 0.25


,5 =05


0.375 6 = 0 875


, =05


06 =0.125


=0


S=-05


1=1




0 = 0.625


= 1 0, =0.75 4, =0.5

(C) (D)

Figure 4-3. 2D primitives and its nodal density distribution A) Quarter circle B) Semi
circle C) Triangle D) Wedge shape


0, = 0 875


0, = 0 375


0(


(1,0.5)


6


3 = O(0,









4.2 Mapping in 2D Solid Elements

The 9-node quadrilateral element has bi-quadratic basis functions as shown in

Table 3-1. The same basis functions could be used to define the mapping between the

parametric space and the real space. However, it can be shown that the same mapping is

obtained when the bilinear basis functions are used if the element is a parallelogram with

straight edges in the real space and the mid nodes are at the center of each edge. In

addition the center node should be located at the center of the element. Using the bilinear

basis function for mapping simplifies inverse mapping from real space to the parametric

space. If the biquadratic basis functions are used, the mapping equations are non-linear

simultaneous equations having multiple solutions in general. The following equations

show the 4-node mapping basis functions.

1 1
M =(1+ r)(1+ s),M, =(1-r)(1+s)
4 4 (4-5)
1 1
M3= 4(1-r)(1-s),M4 =(1+r)(1-s)
4 4
A simple proof is given below to show that the mapping from the real space to the

parametric space using biquadratic basis functions of 9 node quadrilateral elements is

identical to the mapping using bilinear 4 node basis functions if the element has straight

edges with mid nodes at mid points of each edge and the center node at the center of the

element. The mapping equations using biquadratic basis functions are as follows

9
X = M,(r,s)X,
i=1
1 1 1
= (r +r)(s2 +s)X1 + (r2 -r)(s2 +s)X2 + (r2 -r)(s2 -s)X3
4 4 4
1 1 1
+ (r2 +r)(s2 -s)X4 + (1-r2)(s2 +s)X5 + (r -r)( s2)X6 (4-6)
4 2 2
1 1
+ (1-r2)(s2 -s)X7 + (r2 + r)(1- s2)X, + (1-r2)(1- s2)X9
2 2









where Xi is the nodal coordinates at node i and the numbering scheme used

corresponds to Figure 2-1.

The condition that mid-edge nodes are at the mid points of each edge and the center

node is at the centroid of the parallelogram can be stated as follows.


X, + X2 X2 +X3 X3 + X4 X4 + X,
X,5= X6= X,7= X, =
2 2 2 2

X + X2 + X3 + X4 (4
4
Substituting Eq. 4-7 into Eq.4-6 and rearranging yields


1 1 1
X= ZM,(r,s)X = (+ r)(l+s)X + (1- r)(l+s)X2+ (1r)(1-s)X3
4 4 (4-8)
1 4
+ (1+r)(1-s)X4 =ZM (r,s)X1
4 i
The above equation shows that the mapping for the 9 node element is identical to

mapping for the 4 node element when the 9 node element is a parallelogram satisfying

the conditions in Eq. 4-7.

Parametric coordinates corresponding to a given global coordinates are unique for a

4-node quadrilateral element when it is a parallelogram. To make the calculation simple

and fast, an analytical solution for the quadratic 4-node element has been used. But in

general there is no analytical solution for higher order elements such as quadratic 8 or 9

node elements. For such elements, in general, it is necessary to use iterative methods such

as the Newton-Raphson method to find a solution for inverse mapping. However such

numerical computation can be avoided when the higher order elements are

parallelograms because, as shown above, the mapping is identical to that of bilinear

elements when the mid nodes are centered.









The analytic solution for the inverse mapping for 4 node quadratic elements can be

obtained as follows. The mapping equations are:

4
x(r, s)= N, (r,s)x, (4-9)

= 0.25(1+r)(1+s)xl + 0.25(1-r)(+s)x2 + 0.25(1-r)(1-s)x3 + 0.25(1+r)(1-s)x4


4
y(r, s)= N, (r,s)y, (4-10)

=0.25(l+r)(l+s)yl + 0.25(1-r)(l+s)y2 + 0.25(1-r)(1-s)y3 + 0.25(l+r)(1-s)y4
Expanding and rearranging the above equations are be expressed as follows.

4
x(r, s)= N,(r, s)x, = al + b r + C + dirs (4-11)


a, = 0.25(x + x2 + X3 + X4)
where b =0.25(x -x2 -x3 +x4)
c, =0.25(x + x2 -X3 -x4)
d, = 0.25(x, -x2 +X3 -4)
4
y(r, s)= N, (r, s)y, =a2+ b2 r + 2+ d2rs (4-12)


a2 =0.25(y, +y2 +y3 +y4)
where b2 =0.25(y, y2 3 + 4)
where
c2 =0.25(y + y2 -3 4)
d2 =0.25(y y2 + 3 4)
If the element is a parallelogram in real space, its two diagonals bisect each other.

Therefore, di = 0 and d2 = 0. Then the following simple analytical solution as shown in

Table 4-2 can be found.

Table 4-2. Analytical solutions for the 4 node quadratic element mapping (parallelogram)
bl 0 and c2 0 bl =0,b20 andcijl0
y-a x a cbsy 2 x a, blr
y a2 b2 ( 2 2x a S
x-a, cs X a br c,
r- s s- r-
b1 c2 C1 b2












Even if the element is not a parallelogram, an analytical solution for the mapping

from global coordinates to parametric coordinates can be found even though the solution

to the mapping equations is not necessarily unique. Therefore, in this case, the validity of

(r, s) should be performed by checking if the obtained r and s satisfy Eq. 4-6.

As shown in Table 4-3, the mapping for a non-parallelogram element can have

multiple solutions including imaginary solutions. Therefore the 2D element shape in this

system is restricted to parallelogram to avoid such ambiguity in mapping.

Table 4-3. Analytical solutions for the 4 node quadratic element mapping (non-
parallelogram)
A O0 A= 0 and B :0
C-Bs C -Ar
r=--7 s=
A B
B 0 B=0 A 0 A=0
-m+ m2 -4nl s -n -m im2 -4nl r -n
s= r=
21 m 21 m
where m2 4nl > 0 where m2 4nl > 0


A = b2 2 b A = b2 2 b
dB dh
Where < B = c2 2c1 where < B = c2 c,
d, d,
C=y x-a2 1 2 a,
=- .. + 7 = 7 + l
d, ad, ad, d,
l=d,B l = d,A
Where m = bB Ac, -dC where m = cAA bA dC
n = Ax- Aa1 bC n = Bx- Ba1 cC














CHAPTER 5
CONSTANT CROSS-SECTION SWEEP

It is possible to create commonly used primitive solids such as spheres, cylinders

and cones using 3D implicit solid elements. However, the current solid modeling practice

involves creating design features by sweep operations such as extrude, revolve and sweep

along arbitrary curves. While implementing such operations are easy using B-Rep

models, it has not been attempted using implicit surfaces models. In B-Rep models, 2D

profiles are defined as a loop of edges and swept along a curve to create sweep features.

Surfaces generated by sweeping these edges along any curve are easy to define as

parametric surfaces. To accomplish the same using implicit solid elements, the 2D

profiles that are swept to create 3D features can be defined by Boolean combination of

simple 2D primitives defined as described in the previous chapter. To define a 3D sweep

feature using this 2D profile, it is necessary to define 3D implicit solid elements for

sweep operations that are derived from any 2D implicit solid element that has been used

to define the 2D profile. If the profile is defined using Boolean combination of multiple

2D primitives, then a corresponding 3D primitive is created for each 2D primitive. The

3D sweep feature is then obtained by a similar Boolean combination of the 3D primitives.

In the following sections, 3D implicit solid elements are defined for representing solids

obtained by extruding, revolving and sweeping a given 2D primitive where constant

cross-sections along the parametric t direction are obtained. A more general sweep

operation is discussed in the next chapter where variable cross-sections are allowed.









5.1 Extrude Elements

The extrude element is used to represent the solid obtained by extruding (or

sweeping along a straight line) a given 2D primitive defined using an implicit solid

element. The direction of extrusion is assumed to be normal to the plane of the 2D

primitive. By definition, every section of the extruded solid normal to the direction of

extrusion should have a cross-section identical to the 2D primitive being extruded.

Assuming that the 2D primitive to be extruded is defined by an implicit solid element

with n nodes, the density function within the element can be expressed as follows

n
2D (, S)N2(r, s) (5-1)
i=1
The nodal values of density "2D are interpolated within the element using

polynomial basis functions ND (r,s). Figure 5-1 shows the 2D element being extruded

and its corresponding extrude element that has twice the number of nodes and its depth D

in the parametric direction t is the desired depth of extrusion. Each node "i" on the front

face (t=-) of the element and the corresponding node (i+n) on the back face (t=-l) are set

to have equal values of density to ensure that both faces represent the same cross-section

geometry.



Extrusion




Figure 5-1. 2D primitive element and the corresponding extrude element

The density distribution within the extrude element is interpolated as









2n
xt (r, s, t)= ~i3D (5-2)
l=1
The basis functions for interpolating the density within the extrude element are

derived such that the cross sectional shape at any t is identical. The basis function N3D at

the node i for this element can be defined as follows.


N 3D(r,s, t) = (1+ t)N2D for i =.....n (5-3)
2
3D1 It)NZD
N3 (r,s, t)= (1-t)N1 for j=n+i
2
The above basis functions ensures that the density distribution within the element is

independent of t as shown below


ZEx (r,, ,t) = N3DExt = 1 ( + t)j D Ext + 1 t)y N2D Ext
=2 1 2 151 (5-4)

=Z N2D D = 2D (r S)
1=1
where, the condition used is Ext = fExt = 20D

This ensures that every cross section from the front to the back face of the extrude

element is identical to the geometry represented by the 2D primitive element regardless

of the type of the 2D element and the geometry represented by it. The mapping from

parametric space to real space can be defined as follows for the extrude element.

2n
X(r,s, t) = 'MExt (rst)X3D (5-5)
i=1
The mapping has to be linear in t to ensure that the solid represented by the element

is not distorted in the extrusion direction. In general, it is possible to use the basis

functions used for interpolation for mapping also to create an isoparametric element.

However, if the edges of the element are straight lines, the mid-edge nodes are at the

center of the edges and the center node is at the centroid, then regardless of the number of









nodes in the 2D element it is possible to use the following trilinear interpolation functions

Mi (r, s, t) for extrusion.


1 1 1
M1 = (1+r)(l+s)(1+t),M2 =(1-r)(1+s)(1+t),M3 (1-r)(1- s)(1+t)
1 1 1
M4 = (I+r)(1-s)(1+t),M5= (I(+r)(1+s)(l-t),M6= (I-r)(1+s)(l-t) (5-6)

M7 = -(-r)(1-s)(1-t),M, =- (1+r)(1-s)(1-t)
8 8
An examples of a solid created by extrusion are shown in Figure 5-2 where an

ellipse represented using a single 9-node element is extruded to create the solid in Figure

5-2B.



Ir





(A) (B)
Figure 5-2 Solid created by extruding ellipse

The profile to be extruded can be created using multiple elements as shown in

Figure 5-3 where four elements are used. The first primitive was a rectangle from which

an ellipse was subtracted at the bottom. A circle was added to this geometry by union at

the top and another smaller concentric circle was subtracted. Upon extrusion, a 3D

primitive is created for each of the 2D primitive used to define the profile. These

primitives are then combined using the same Boolean operations that were used in the 2D

profile to create the solid in Figure 5-3B.



















Figure 5-3 Extrusion of profile defined using multiple elements

5.2 Revolve Elements

The revolve element is an implicit solid element used to represent the solid

obtained by revolving (or sweeping along an arc) a given profile defined using a 2D

implicit solid element. The revolved solid element is defined by specifying the 2D

element, an axis of revolution and an angle of revolution. It is convenient to use a

cylindrical coordinate system to define the nodal coordinates of the element as

C, = {R, 0, Z, }. The cylindrical coordinate system is chosen such that its z-axis is along

the axis of revolution of the element. The radial coordinates of the centroid of 2D

primitive element is R, as shown in Figure 5-4.






R r

R0


Figure 5-4. Cylindrical coordinate system for revolving

As in the extrude element, the density distribution at the front and back element

faces as well as any cross-section in between should be identical. Therefore, the









interpolation functions used in the extrude element is also used here to interpolate the

density within the revolve element as

2n
Rev (rS,t) = NDRe (5-7)
1i=1
where, ORev are the densities at the nodes of the revolved element and N3D are the

basis functions defined in Eq 5-3. The mapping between the parametric space and the real

space can be expressed as:

2n
C(r, s, t)= C,M, (r, s, t) (5-8)
z=1
In the above equation, C = (R, Z) is the coordinates of an arbitrary point in the

element and Ci are the nodal coordinates both with respect to a cylindrical coordinate

system. The mapping basis functions M, could be the same as the interpolation basis

functions or when the 2D element's edges are straight lines, one could use the trilinear

interpolations functions given in Table 3-2. To express the geometry with respect to a

Cartesian coordinate system yet another mapping is required as illustrated in Figure 5-5.

Let C = (R,0,Z) be the coordinates of a point in the cylindrical coordinate system and X =

(X, Y, Z) be the coordinates of this point in the global coordinate system.

c 0, Z)






Sx--^Xo (ax b


Figure 5-5 Mapping from cylindrical to Cartesian coordinates









The global coordinates with respect to the Cartesian coordinates can be expressed

as:

X(r, s, t) = X +R(r, s, t) cos(G(r,s, t))a R(r, s, t) sin(9(r,s,t))(a x b) + Z(r,s, t)b (5-9)
In the above equation, Xo is the position vector of the origin of the cylindrical

coordinate system while a and b are vectors along the radial and axial directions

respectively of the cylindrical coordinate system.










Figure 5-6. Examples for revolved solids

Two examples of solids created by revolved elements are shown in Figure 5-6,

where a sphere is constructed by revolving a semi-circle. The second solid is created by

revolving a profile defined using three elements. Two circles are subtracted from a square

to create the profile which is revolved 2100 about an axis that is parallel to the edges of

the square.

5.3 Sweep Elements

The sweep element is used to represent a solid obtained by sweeping a profile

represented by a 2D implicit solid element along an arbitrary parametric curve.

Therefore, the sweep element is defined by specifying the 2D element, a sweep trajectory

R(t) as well as an orientation vector d(t) that controls the orientation of the swept profile.

The sweep trajectories can be any kind of parametric curves including composite curves.

If the trajectory is a composite curve, the number of the 3D elements required to define

the swept solid is equal to the number of curves used in the composite curve. The basis









functions, NA,, defined in Eq. 5-3 can be used for this element also to ensure that the

density distribution within the element is independent of the parameter t, so that the

cross-sections will be identical at any value of t. Therefore, the density distribution in a

sweep element can be expressed as

2n
,s" (r, s, t) = N3D (5-10)

v d




E> R(t) Sweep trajectory
z Ro u



x

Figure 5-7 Relationship between the global and local coordinate systems

For the sweep element also it is convenient to use an intermediate (or local)

coordinate system as shown in Fig. 4-7. The local coordinate system is defined using

three unit vectors i, ii and fn. The vector i is tangent to the sweep trajectory can be

computed as the derivative of the parametric curve equation R(t).


= where tdR(t)
|Rt dt

Ixd
n -
li_
lxd (5-11)

m =nxl
If d is invariant along the sweep trajectory, the sweep will be not be twisted. On the

other hand, when d is a function of t and varies along the sweep trajectory, a twisted










sweep will be obtained. In order to create a twisted sweep the user must specify d(t). It is

important to ensure that d is not parallel to the curve anywhere along the trajectory.

The nodal coordinates of the sweep element is represented with respect to the local

coordinate system as L(r,s,t)= {u(r,s) v(r,s) t}T The mapping between the parametric

space and the local coordinates u and v can be expressed as

n
u(r,s)= u,ND (r, s)
I=1
(5-12)
v(r,s) = vN2D (r, s)
1=1
where, ui, vi are nodal coordinates of the 2D element with respect to the local

coordinate system. The mapping between parametric and the global coordinates can

therefore be expressed as:


X(r,s,t) =R(t)+ fu + iiv or

X(r,s,t) =R(t) + fi uND (r,s) + m 'N2D (r,s) (5-13)
i=1 i=1
Figure 5-8 shows two examples created using sweep elements. In the first example,

a circle is swept along a cubic spline composed of three Hermite curves. Three sweep

elements are therefore used, one for each curve of this composite curve. In the second

example a more complex profile defined using nine 2D element is swept along an

arbitrary composite curve. The resultant solid is obtained by Boolean combination of the

solids represented by individual elements.






rf~=


'p


Figure 5-8 Examples of solids created using sweep elements














CHAPTER 6
VARIABLE CROSS SECTION SWEEP

As mentioned in Chapter 5, a more general sweep operation can have variable

cross-sections. Therefore the variable cross-section sweep using blend solid elements are

described in this chapter.

Blend elements are used to represent solids created by blending between a pair of

2D primitives (or cross-sections) represented by 2D implicit solid elements. The 2D

implicit solid elements used for representing the 2D primitives to be blended are referred

to as "2D profile elements" in the following. Since a blended solid can be generated from

two or more 2D profile elements, the total number of blend elements used for generating

a blended solid depends upon the number of 2D profile elements. In other words, ifm 2D

profile elements are given then (m-1) 3D blend elements are required to create a solid

primitive that blends between these 2D profiles. Unlike extrude, revolve and sweep

elements, the shape of the cross-section can vary from one profile to another profile along

the t direction in the parametric coordinates. Therefore the intermediate shape from one

profile to the next has been studied to obtain a clear understanding of the nature of the

blended solid obtained.

Two different types of blended solid primitives have been defined in this chapter,

namely, straight blend and smooth blend. This classification is based on the surface

continuity between the solids defined in adjacent elements. Straight blend is Co

continuous while smooth blend defines a C1 continuous blend so that the surface of the

primitive defined by these elements is tangent continuous at the interface between









neighboring elements. Blend elements also can be classified based on the sweep

trajectory along which the 2D profiles are swept. If the sweep trajectory is a straight line,

it is called "extrude blend element" simply denoted by "blend element". If the sweep

trajectory is a general parametric curve, it is called "sweep blend element".

To define blend elements, all 2D profile elements used for representing 2D profiles

should be the same type of element. In other words, all 2D profile elements should have

the same number of nodes per element and the same shape function for each node but

they may or may not have the same size and nodal density distributions. In the following

section, the detailed description of the blend elements is described.

6.1 Straight Blend Element

Two 2D profile elements and the distance between them are required to define a

straight blend element. Let the shape density functions of front and back 2D profile

elements be "F2D and "B2D respectively.


F2D = 2D (r, )F =N2DF + NDF + ..... (6-1)
1=1

~B2D =YN,2DrS)OB
1 = N,2D D + NDA +. + ND (6-2)
N2D is the 2D basis function at the node i and n is the number of nodes per

element in the 2D profile element.

Now, density distribution within the straight blend element can be defined by the

following shape density function.


STBIend (r, t) = Y N3D, (6-3)
1=1









The basis functions for interpolating the nodal densities are designed to give linear

interpolation in the parametric direction t that is the desired direction of depth. The

following equation shows the basis function N3D at the node i.


N3D (r, s, t) = (1 + t)N,2D (r, s) for i = 1.... n at the front face of the element
2

ND (r,s, t) = (1 t)N2D (r, ) forj= n +i at the back face of the element (6-4)

The mapping from the parametric space to the real space for the straight blend

element can be defined as

2n
X(r,s,t) = M (r,t) (6-5)
=1-
XD and M,3 are the nodal coordinates and the mapping basis function at the node

i respectively. As in the previous chapter, it is assumed that the mid-edge nodes are at the

mid point of the edges and that the center node is at the centroid so that the trilinear

mapping functions defined in Eq. 5-6 can be used.

The cross-sectional shape at the common face between elements is identical to the

2D primitive used for creating the straight blend element. Therefore the blended solid

defined by the straight blend element has the following properties.

Property 6.1-1: The blended solid represented using the straight blend elements

satisfies Co continuity at the interface between each neighboring element.

















t = 1 at
Element B


t = 1 at Neighboring face
Element A [t= -1 at Element A, t= 1 at Element B]

Figure 6-1. Two neighboring straight blend elements having quadrateral 9 node elements
as 2D profile

Proof: It can be verified by showing that the density distributions at t = -1 at the

element A and t = 1 at the element B are identical to the density distribution of the 2D

profile element used for the common face. Therefore the shape of interest at the common

face between two neighboring elements is same as the 2D profile used.



nd (r, ) N2D (r, S)B 2D at the element A
i=1

mTBlend (, s,) N2D (rs)F= F 2D at the element B (6-6)
'=\
B2D = F2D because the same profile element is used for representing the back

face of the element A and the front face of the element B.

As mentioned earlier, the 2D profile elements may or may not have the same

density distribution. If the density distributions of the front and back 2D profile elements

are identical, the surfaces connecting the front and back 2D profiles have the following

property.









Property 6.1-2. The surface connecting the front and the back 2D profiles in the

straight blend element is composed of straight lines that connect points having the same r

and s but different t value (i.e. t = and t 1 respectively) only ifF = eB.


X(r*, s*, 1





-. X(r*, s*, -1)


Back face where t = -1
Front face where t=1
X(r*, s*, t)

Figure 6-2. A straight blend element having quadratic 9 node elements as 2D profiles

Proof: If the density distribution for the front and the back 2D profile elements are

the same, the shape density function for the straight blend element can be obtained by

applying the condition = ,F _= B as follows.


STBlend N2 (r, s)O, (1 + t) + (1 t) = N2 (r, s)q, (6-7)
12 2 1
Eq. 6-7 shows that the cross-sectional shapes along the parametric direction t are

invariant with respect to the t in the parametric space. Suppose (r*,s*) is the parametric

coordinates of a point on the boundary so that 2D (r*, s) = b (boundary density) then

X(r*, s t) is the position vector of a point on the curve along the surface of the solid

connecting points on the front and back faces that share the same parametric coordinates

(r*,s*). It would now be shown that this curve is a straight line.













1- r1 ,s2 1z 1 2,s) (1t)


N 2D F 2Dz NBr
1=1 1





In the above equation, (x[, y, z[) and (xS, yB, z ) are the nodal coordinates at
1 1



the front and the back face of the straight blend element respectively and XF and XB are


defined as



1 1
N2D *T S + t)z *\2D S)( t)zT







2 2







N rs )x., N7(r ,s )y,
1-i 1-i
( (+ t)XF + -- ( t)XB (6-8)
























points having the same r and s and different t 1 and t 1 respectively. Such a surface



clear that in general when the 2D profile elements defining the blend element have
N,2D*S)F YN'2D S
1=1 (=1 s x-
















diretion, the following property can b e obtain the point Xd.and X From Eq. 6-7
the density values at XF and XB are same and the densities through the line is also same.

Therefore it is verified that the surface is composed of straight lines connecting two

points having the same r and s and different t = -1 and t = 1 respectively. Such a surface

is known as a lofted surface (or ruled surface) [Filip and Ball, 1989]. Intuitively, it is

clear that in general when the 2D profile elements defining the blend element have

different density distributions the blending surfaces are not guaranteed to be lofted

surfaces.

Investigating cross section shapes of the blend element along the parametric t

direction, the following property can be obtained.










Property 6.1-3. The cross-sectional shape of the straight blend element at a

certain parameter t* is a parallelogram if the mid-edge nodes are at the center of each

edge and mid-face node is at the centroid of the face.

s=1 6

r=-1\ X(-1,1, t*) s=l X(1,1, t*)


5 -X r= 1
r r=


3- X(-1,-1, t*) s = -1 X(1,-1, t*)

t=1 4 t=t* t=-I

s = -1 r = 1


Figure 6-3: Cross section at a certain t in the straight blend element

Proof: To prove that the cross-section is a parallelogram it is first shown that the

edges of the cross-section are straight lines. It can be assumed that the mapping basis

functions are tri-linear regardless of the number of nodes in the element if the mid-edge

nodes are at midpoint of the edges and mid face nodes are at the centroid of the face.

Therefore the mapping equation can be expressed as

8
X(r, s, t) = [ M,(r, s, t)X, (6-10)
1=i
In the above equations Mi are the trilinear basis functions and X, are the nodal


coordinate points in the global coordinate system. Expanding the above equation by


substituting the trilinear basis functions for Mi(r,s,t), yields,









1 1 1
X= (1 + r)(1 + s)(1 + t)X, + (1- r)(1 + s)(1 + t)X2 + (1- r)(1- )(1 +t)X3
8 8 8
1 1 1
+ -(1 + r)(1 s)(1 + t)X4 +l(l+ r)( + s)(1- t)X, + (1 -r)( +s)(1- t)X6 (6-11)
8 8 8
1 1
+ (1- r)( s)( t)X, + (1 + r)(1- s)(1- t)X,
8 8
At any cross-section of the element t is constant so it will be denote as t*. The

equation of the boundaries of the element at this cross-section can be obtained by setting

parametric coordinates r and s to -1 or 1.

Consider the edge where r = 1, the equation of the edge, X(1,s,t*), can be expressed

as

1 1 1
X(1,s, t*) = (1+ s)(1+ t*)Xl + (1- s)(1+ t*)X4 + (1 + )(1- t*)X,
(6-12)
1
+ (1 s)(1 t*)X,
4
Rearranging,


X(1,s,t*)= (I+s) ((1+t*)X, +-(1-t*)X +-(1I-s){ (1+t*)X4 +-(1-t*)X,

Since t*, X1,X5,X4 and X8 are constants, it is clear that the equation of X is that of a

straight line where only s is a variable. Similarly it can be verified that other three edges

defined by r = 1, s = -1 and s = 1 are also straight lines. Now to show that the

quadrilateral defined by these four straight lines is a parallelogram it needs to be verified

that the diagonal of this quadrilateral bisect each other.

Let four vertices of the cross-section of the element at a given t* be X1, X2, X3 and

X4 respectively.


1 1
X(1,1,t*) X = (1+ t*)X, + (1- t)X,
x2 2+
1 1
X(- 1,1,t*)= X2 = (1+t*)X2 +-(1 t*)X6
2 2









X(-1,- 1,)= X (l+t )X3 +(l-t )X
2 2
1 1
X(1,- ,t*) = -X4 =(1+ t*)X4 +-(1- t*)X, (6-13)
2 2
The mid point of each diagonal can be represented by

X +X3 (t(1+*)(X,+X3)+ (1-t)(X,+X,)
2 212 2
X +X 1 1 1
2 4 (1+ t*)(X2 + X4) +- (1- t*)(X6 + X) (6-14)
2 2L2 2
From the fact that 2D profile elements are parallelograms, the following equations

can be obtained.

X, + X34 X + X X + X X +8 (6-15)
2 2 2 2
Applying the above conditions to Eq.6-14, the following relationship can be

obtained.

X* +X X +X
1 3 2 4 (6-16)
2 2
Therefore the midpoints of the two diagonals are the same thus proving that the

cross-section is a parallelogram.

This is a very important property because it can be used for the mapping from the

global coordinates to the parametric coordinates. Once it is verified that the cross-section

shape of the straight blended element at a certain parametric coordinate t* is a

parallelogram, then two-dimensional mapping techniques can be used for finding the

parametric coordinates r and s.

Examples of solids created by blending 2D profiles are shown in the following

figure where four 2D profiles are used for both examples. Figure 6-4A shows the









blending between circular shapes and figure 6-4B shows the blending between circular

and rectangular shapes.











(A) (B)

Figure 6-4. Examples of straight blend using blend elements

6.2 Smooth Blend Element

Smooth blend elements construct a C continuous blended solid between 2D

profile elements defined along a straight line. The shape density function for the smooth

blend element can be defined by

2n
end (r, s,t) = (N, + ) (6-17)

q and are the nodal density and its corresponding tangent component at the

node i respectively. The basis functions for interpolating the nodal densities are designed

to give Hermite interpolation in the parametric direction t. The following shows the basis

function N3D and N3D at the node i.



2 2=
D H 22DH for i =1.....n


N,3D N,2D +1 t1 =2DH2 for i= .....n (6-18)
2 )










3D N 2D 1-3 t 2 2 3 2DH3 for j=n+i



N 3D 2N2D t 2 +l2 + 2DH 4 O 1 3
VN = 2t + =ND H4 for j = n +i

A composite Hermite curve is guaranteed to be C1 continuous regardless of the

tangent at the nodes. However, to set the tangent components at each node automatically,

the following two additional conditions are used. First, the second derivative at the end

each element should be equal to that at the beginning of the adjacent element. Second, the

second-order derivatives are set to zero at both ends of the primitive. According to these

conditions, the following linear equations are obtained, by which an automatic method

for setting the tangent components can be achieved. The following equation is a matrix

form for N simultaneous linear equations. Therefore, the tangent components can be

computed by solving the following equation.


2 1 0 0 ........ 0 R 3R2 -3R1
1 4 1 0 : R' 3R -3R1
0 1 4 1 : R3 3R4 3R2




0
*= (6-19)



1 4 10
0 1 4 1 RN1 3RN -3RN2
0 0 0 1 2 RN 3R, -3RN
In the above equation, R1 = 4 and R' = In the smooth blend element, the

mapping from the parametric space to the real space can be performed as

2n
X(r, s, t)= (MD (r, s,t)X3D + MAD (r, ,t)X3D) (6-20)
i=1










XD and XtD are nodal coordinates and its tangent components at the node i.


M,3D and M2D are mapping basis functions that are identical to N,3D and N3D


respectively for this element. X3D is computed from Eq. 6-19 setting R1 = X3D and


R' = XD The following property is proved to show that the solid created using the

smooth blend elements has C1 continuity along the parametric coordinate t.

Property 6.2-1. The blended solid using the smooth blend elements satisfies C1

continuity at the interface between adjacent elements.

VO (r, s,- 1) = -V, (r, s,1)






Element
SEl n Element B
Element A -.


t = 1 at Common face
Element A [t= -1 at Element A, t= 1 at Element B]


Figure 6-5. Two neighboring smooth blend elements having quadratic 9 node elements as
2D profile

Proof: This property can be verified by showing that the surface normal vector is

continuous at the interface between each neighboring element. Suppose the face of

interest is the back face (i.e. t = -1) in the element A denoted by EA and the front face

(i.e. t = 1) in the element B denoted by EB as shown in figure 6-5. The surface normal

vector is computed by gradient of the shape density function or normal to the element

surface in this system. The following is the formula for computing the gradient of the

shape density function with respect to the global coordinates x, y, z.









00 00 9ax ay az
ax ar dr dr dr
a [ a ax (6-2z
V4 = < = [j] > Where, the Jacobian matrix [J]= x az (6-21)
ay as as as as
9 00 9ax ay az
z, at a 9t at at
Let the shape density function at the element A and the element B be EA and 4EB

respectively, then


(r, s, t) = ND (r, s)[H (t)q, +H2 (t)Ah + H3 (t) A, + H4 (t)q4 ]
1=1

EB (r, s, t) = N,2D (r, s)[H (t), + H2 (t) + H3 (t) + H4 (tB] (6-22)
i=1
Since both element A and element B are sharing the 2D profile element at the

common face, the density distributions at the back face of the element A and the front

face of the element B are identical. Therefore,


S= = and = = (6-23)
Differentiating 4EA and 4EB with respect to r, s and t and thereafter substituting -1

for the parameter t at the element A and 1 for the parameter t at the element B into Eq. 6-

22 and applying Eq. 6-21.


S(r, ,-1)2D (rs) EB (r, s,1)
ar a dr dr

S(, 1) (r, s) 0 EB (r, s,1)


0_EA, (r, 1) 1 0EB (r, s,,1)
-)= N2D (rs = (6-24)
at 2 ,=7 at
Similarly, the above procedure for x, y and z instead of 4 can be applied.

Now, let x, y and z be xEA y z, and xEB EB ZEB with respect to the

corresponding elements. Then











=1

EB (r, s, t)= N2D (r, s)[H (t)xB + H2 (t)B + H3 (t)x + H4 (t)xB] (6-25)
I=1
Where x = xB = xF and x,= xB, = xF,

Differentiating xA and xEB with respect to r, s and t and then substituting -1 for the

parameter t at the element A and 1 for the parameter t at the element B into Eq. 6-25.


xEA (r, s,- 1) n aN2D (r, s) XEB (r, s,1)
Or 1 = r r

xEA (rs, s-1) N2) (r, s) xEB (r, s,)
as i=1 s s

axEA (r, S,-) n 2DEB (r, S,1)
N,2D (r s)x, B( 1) (6-26)
at 2 at
Following the same procedure for y and z, the similar relation to Eq. 6-26 for y and

z can be found.

From Eq.s 6-24 and 6-26, yields,


VE1(r, S,-1)= V0E2, (r,,1) (6-27)
Therefore Eq. 6-27 proves the property 6.2-1.

If the density distributions between the front and the back 2D profile elements are

identical, the surfaces connecting the front and back 2D profiles have the following

property.

Property 6.2-2. The surfaces connecting the front and the back 2D profiles in the

smooth blend element are composed ofHermite curves that connect points having the

same r and s but different t value (i.e. t = -1 and t =1 respectively) only ifF = B .



















r Back face where t = -1
Front face where t=1
X(r*, s*, t)

Figure 6-6. A smooth blend element having quadratic 9 node elements as 2D profile

Proof: If the density distributions for the front and the back 2D profile elements are

same (i.e. OF = 0B ), it implies that all tangential components, are zeros according to

Eq.6-22. Therefore the density function can be simplified as follows.


~SMBlend n 2D + 1)2 2D t + 1 2 t + 1 )3
Ze N, =(r,s)r 3t 2 t+ J + N,2D (r,s) 1 3 + 2 1]B
1=1 2 =1 2
Applying the condition = 0F = 0B, then

SMsBIen = N2D (-r, S) (6-28)
i=l
The above equation shows that the cross-sections along the direction of t in the

parametric coordinate system are invariant. Let (r*, s*) be the parametric coordinate of a

point on the boundary of the solid so that 2D (r*, s*) = b (boundary density), then density

value at every point on a straight line connecting (r*, s*, -1) and (r*, s*, 1) in the

parametric coordinate system is Ob because Eq. 6-28 shows that density is invariant with

respect to t. Therefore, this line is guaranteed to be on the surface of the solid. The

parametric equation of the line mentioned above in the global coordinate system is













S I=1
X(r*, s*, t) =y IN N2D*,S')H(t) yF + H2(tytF+H3 ( + H4 h

N,2D(r *, s*)[H(t)z7 + H2(t)zF + H3(t)zf + H4(t)z]
I=1

=H,(t)XF + H2()XF +H3(t)XB + H4(t)X
This is a Hermite curve equation where



N7 D(r ,., ), c -N, (r ).
n n


X F and

IN ( ) D i, ND(r*,s*)z[
XF__ F __ 2D *S)y > and XF tB >2D S(* *)tF
F f =1 B 1=1
z n (rZ t n (r,



XB Nz2D *,s)y and X F S Z ND *, S )yF
S 1N2 D(r,s)z (r,s)z



1 1
n n
Sc i 2D *,t s s te f 2D l*,g e tiB
Xt IX X t =
n n
XWhe = re = < N2 *,S) and XX = Fy = h 2D *,SXF
T B e2 D=1 B 2=1
N,2D *, a S b1 wt a

This curve is also a cardinal spline if it satisfies the following equation.


XF +4X, +X,, =3(X XF)
Where X, = X E = XF


This equation can be written as follows.


N (r ,s )'tEA +4xh + XEB
2=1

S N r2D *\i1F A B
N2Dr ,s ) zEA 4 +4yz +, EB


2=1
N 2D *,S F +4z, +zB,


ZN72D BS XB


N D *,S ) Bz
-=1


-=1


-i1


Or


N x2D + 4X, +X = 3 N2D *r, )( X XF)
2=1 2=1


(6-29)







(6-30)








(6-31)







(6-32)


F}



ZEA
ZF }
Z EA,


(6-33)










+ t,= 3 B (XB F)
Since the solid element is defined such that X h + 4X1 + X (B = 3(XB XEA


(see Eq. 6-19), the above equation is automatically correct. Therefore when density

distribution is identical for front and back faces the smooth blend primitives have C2

continuity along the t direction at the common face between neighboring solid elements

because curves on its surface in the parametric direction t are Cardinal splines.

However, in general when 2D profile elements defining the smooth blend element

have different density distributions, the resulting blended solid's boundary is not

composed of cardinal splines.

Investigating cross-sectional shapes of the smooth blend element along the

parametric direction t, the following property can be obtained.

Property 6.2-3. The cross-section shape of the smooth blend element at a certain

parameter t* is a parallelogram if the mid-edge nodes are at the center of each edge and

mid-face node is at the centroid of the face.


6
10 t = -1 at element B
I 9 X(-1,1, t*) s = 1 X(1,1, t*)


12 r=-1 r=

3X(-1,-1, t8) s = -1 X(1,-1, t*)


A) t=t*
t = 1 at 1 Common face
element A [t= -1 at element A, t= 1 at element B]


Figure 6-7. Cross section at a certain t* in smooth blend elements









Proof: To prove this property, it is first verified that it is true in the smooth blend

element. Initially it is assumed that 4 node quadrilateral elements are used for the 2D

profile elements. The mapping equation as shown in figure 6-7 is

8
X(r, s, t) = (N, (r,s, t)X, + N,(r, s,t)X,) (6-34)
'=1
Consider the edge where r = 1, for any cross-section t=t*. The points on this edge,

X, can be written as

1 1
X(1, s, t)= (1 + s)H, (t)X + -(1- s)H (t)X4
2 2
1 1
+ (1+ s)H3(t)X, + (1- s)H3 ()X
2 2 (6-35)
1 1
+ (1+ s)H2(t)X,, + (1- s)H2 ()X,4
2 2
1 1
+ (1 + s)H4(t)X, + (1- s)H4(t)X,
2 2
From Eq. 6-19, it is known that,


2Xt, + X,5 = 3X5 3X,

2Xt4 + X,, = 3X, 3X4

Xt + 4X5, + Xt9 =3X9 3X1

Xt4 +4X8, + Xt2 = 3X12 3X4
According to the above relationships, X(1,s,t) can be simplified as


X(1, s, t)= (1+ s)(H, (t)X, + H2 (t)X, + H3 (t)X + H4 ()X,,5
2 (6-36)

+ 1(1- s){H1 (t)X4 + H2(t)X,4 + H3 ()X, + H4(t)X,, }
2
Since t, Xi, X5, X4, Xs, Xtl, Xts, Xt4 and Xts are constants, therefore the final form

of X(1,s,t) is a line equation where the variable is s. Similarly it can be verified that other

three edges defined by r = 1, s = -1 and s = 1 are also straight lines. This proves that the









cross sectional shape of the blend element at a certain t* is a quadrilateral. This

quadrilateral is a parallelogram if its diagonals bisect each other as shown below.

Let four vertices of the cross-sectional shape be X,1, X X and X4 respectively.


X(11, t) = x = H, (t)X, + H (t)X,, + H3 (t)X, + H (t)X,

X(-1,1,*) = X = H(t)X2 + H2(t)Xt2 + H3(t)X + H(t)Xt6

X(-l,- 1, t) = X* = H,(t)X3 + H2(t)Xt3 + H3(t)X, + H4(t)X,,

X(1, -1,t) = X4 = H (t)X4 + H2 (t)X4 + H3 ()X, + H4 ()X,, (6-37)
The midpoints of the two diagonals are


1 +3 -1 [H,(t)(X + X3) + H2(t)(X,, + X3)
2 2
+ H3 (t)(X, + X7) + H4 (t)(Xt5 + X,7)

X 2 4 _1 [H(t)(X2 + X4)+ H2(t)(Xt2 + Xt4)
2 2 (6-38)
+ H3(t)(X6 + X,) + H4(t)(Xt6 + Xt)
Again from Eq. 6-19, the followings can be obtained.

2Xt, + X5, = 3X, 3X,

2Xt2 + Xt6 = 3X6 3X2

2Xt3 + X,7 = 3X7 3X3

2Xt4 + Xt = 3X, 3X4
The above equations can be combined to yield the following equations.
2(X,, + Xt3) + (X5, + X,7) = 3(X, + X) 3(X, + X3)

2(Xt2 + Xt4) + (Xt6 + X8,) = 3(X6 + X,) 3(X2 + X4) (6-39)
From the fact that the 2D profile elements used to define the smooth blend are

parallelograms,

X + X3 X2 +X4 and X + X X6 + X(6-40)
and (6-40)
2 2 2 2
Therefore, the following equation can be obtained from Eq. 6-39.









2(X,, + X,3) + (X5, + X,7)= 2(Xt2 + Xt4) + (Xt6 + X,8) (6-41)
The following is also satisfied.

X,, + 4Xt + Xt9 = 3X9 3X,

Xt2 + 4Xt6 + Xto = 3X10 3X2

Xt3 + 4X7, + Xt = 3X, 3X3

Xt4 + 4X8, + Xt2 = 3X2 3X4

The above equations can be combined and yield the following equations.

(X, + Xt3) + 4(Xt, + X7,)+ (Xt9 + Xt,) = 3(X9 + X,) 3(X + X3)

(Xt2 + X4) + 4(Xt6 + X,,) + (Xo + Xtl2) = 3(X,1 + X12)- 3(X2 + X4) (6-42)
From the fact that 2D profile elements are parallelograms,

X, + X3 X2 + X4 and X9 + X1 X1 + X12 (6-43)
2 2 2 2
Therefore, the following equation from Eq. 6-42 and 6-43 can be obtained.


(Xt, + Xt3) + 4(X,, + X7,) + (Xt9 + X,,t) =
(Xt2 + Xt4) + 4(Xt6 + X,,) + (Xt0o + Xt12)
Applying Eq. 6-41 and Eq. 6-44 to the Eq. 6-38, yields

X* + X* X2 +X*
1 3 2 4 (6-45)
2 2
Therefore the cross-section is a parallelogram since Eq. 6-47 shows that the

midpoints of the two diagonals are coincident.

So far, it is assumed that 4 node elements are used as 2D profile elements to verify

property 6.2-3. This is also true when 9 node elements or others are used if their mapping

equations are equal to the mapping equation at 4 node elements as explained below. Let

the mapping equation of the smooth blend element having 4 node profile elements be X4.









4
X4 (r, ,t)= (HN, (r, s)X, + HN, (r, s)X,)
1=1 (6-46)
+ Z(HN (r, s)X+4 + H4N (r, s)Xt(,+4))
1=1
The following shows the mapping function at the smooth blend element where

quadratic elements having n nodes are used for 2D profile elements.


X" (r, s, t)= (H1N, (r, s)X, + HN, (r, s)X,)
1 (6-47)
+ (H,(N, (r, )X + H)X,,, n))

If the mapping equation in the 2D profile element having n nodes is same as the

mapping equation in the 4 node quadrilateral element, i.e., if

n 4
N, (r, s)X, = N (r, s)X, (as proved in Chapter 4 for 9 node elements, Eq. 4-8), the
1=1 1=1

following is also true.


X4(r,s,t)=X"(r,s,t) (6-48)
This property can be used for the mapping from the global coordinates to the

parametric coordinates. Once it is verified that the cross-section shape of the smooth

blend element at a certain parametric coordinate t* is a parallelogram, then two-

dimensional mapping techniques can be used for finding the parametric coordinates r and

s.

Examples of solids created by blending 2D profiles are shown in the following

figure where four 2D profiles are used for both examples. Figure 6-8A shows the

blending between circular shapes and figure 6-8B shows the blending between circular

and rectangular shapes.

























(A) (B)

Figure 6-8. Examples of Smooth blend using blend elements

6.3 Sweep Blend

The sweep blend element is used to represent a solid obtained by sweeping 2D

profiles represented by 2D implicit solid elements along an arbitrary parametric curve.

Therefore, the sweep blend element is defined by specifying the 2D profile elements, a

sweep trajectory R(t) as well as an orientation vector d(t) that controls the orientation

of the swept profile. The detailed description regarding sweep trajectories is explained in

Chapter 5.

Two different types of blending functions, straight and smooth blending can be

applied to sweep blend too. Therefore all properties illustrated already in the previous

sections are also applicable to sweep blend.

Since the nodal coordinates of the sweep blend elements are represented with

respect to the local coordinate system as L(r,s,t) ={u(r,s,t) v(r,s,t) t}T, the mapping

between the parametric coordinates and the local coordinates u and v can be expressed as

n
u(r,s,t)= u,N,(r,s,t)
t1=










v(r,s, t)= vN,(r, s, t) (6-49)

where, ui, vi are nodal coordinates of the sweep solid element with respect to the

local coordinate system. Ni(r,s,t) are basis functions for both the mapping from the

parametric coordinates r, s and t and interpolating nodal densities. Therefore the basis

functions are different with respect to the blend types such as straight blend and smooth

blend.

Examples of solids created using sweep blend element are shown in the following

figure where four 2D profiles are used for both examples. Figure 6-9A shows the sweep

straight blending between rectangular shapes and figure 6-9B shows the sweep smooth

blending between circular shapes.














(A) (B)

Figure 6-9. Examples of Sweep Blending A) Sweep straight blend B) Sweep

smooth blend














CHAPTER 7
CSG REPRESENTATION USING IMPLICIT ELEMENTS

CSG representation scheme is used for representing a solid in this solid modeling

system where implicit solid elements are used for representing primitives instead of half

spaces. The final solid is constructed by combining simple primitives using regularized

Boolean operations to define a solid as a closed region in space. It is necessary to

compute points and surface normal vectors on the boundary of the solid for graphically

displaying it. The following sections, describe the CSG tree structure used in this system,

how to compute surface normal vectors, and how to perform set-membership

classification.

7.1 Surface Normal Vectors

To graphically display primitives defined using implicit solid elements, it is

necessary to evaluate points on the surface and to compute surface normal vectors at

these points. The method for evaluating points on the surface is explained in chapter 9.

Here, the method for computing the surface normal vector at a given point on a primitive

represented by an implicit solid element is described.

To compute the surface normal vector at a given point on the boundary of a solid

represented using an implicit solid element, the point has to be first classified as inside

the element (4 = k1b,-1 < r < 1, -1 < s <1,-1 < t < 1) or on the boundary of the element

(r = +1 or s= 1 or t= 1). If the point is inside the implicit solid element, the gradient of

the density function is computed since it is a vector normal to the surface at the point of









interest. The gradient of 4 with respect to the global coordinates (x, y, z) can be computed

as:



Ox Or
V = -, [J] 1(7-1)
ay ds

az at
W r l) COz
ox ay az
Or Or Or
Where [J]= C- Oz (7-2)
as as as
x ay aoz
at at at
The [J] matrix or the Jacobian matrix can be easily obtained since the mapping

expresses x, y and z as functions of r, s and t. For instance, the first component of the

n
Jacobian matrix can be expressed as where x = xN, (r, s, t).
t=1



S_ rst)_ (xN,(r,s,t)) + a(x2N(r,s,t)) 8a(xnN (r,s,t)) (7
+ +--+ (7-3)
Or Or Or Or Or
Similarly, the other components of the first column are



x N (r, s,t) (xN,(r,s,t)) 8(x2N2,(r, s,t)) O (x N (r,s,t))
as as as as as

x x, (r, s, t) (xN,(r,s,t)) 8(xN2 (r,s,t)) 8(xN (r,s,t))
+ +--+ (7-4)
at at at at at
The second and third columns of the Jacobian matrix can be computed by replacing

x with y and z in Eq. (7-3) to (7-4) respectively.









Note that the gradient points towards the direction of increasing density 4 that is

towards the inside of the solid. Therefore, the surface normal is set in the opposite

direction as n = -V4 .

The actual equations for finding each component of the Jacobian matrix depend

upon the mapping functions of r, s and t. Therefore, the solid elements having different

mapping functions need to be treated accordingly. Since extrude, revolve and sweep

elements have their own mapping functions that are different from each other, they have

their own corresponding equations for finding the components of the Jacobian matrix as

follows.

In extrude elements including extrude blend elements, Eq. 7-3 and Eq. 7-4 can be

used directly because the global coordinates (x, y, z) are just functions of the parametric

coordinates (r, s, t). Therefore each component of the Jacobian matrix can be computed

by just differentiating the mapping function with respect to the parametric coordinates (r,

s, t). However revolve elements use the cylindrical coordinate system whereas the

extrude elements use the Cartesian coordinates system. Therefore one more step is

required to find the surface normal vectors for revolve elements. The following is an

example where the first components of the Jacobian matrix is computed.

cx cx cR Ox cO 8x 8Z
= --+--+ -- (7-5)
Or OR Or Oc Or OZ Or
Here -, and can be derived from Eq. 5-9 explicitly.
OR 00 OZ

Ox Ox Ox
-=a cosO-c sin = -aRsinO-c RcosO and b (7-6)
OR 0o0 OZ
Since R, 0 and Z are represented by the interpolation using the nodal basis

functions, their differentiations are












Or Or






Z a Z,N,(r,s,t)

Or Or
In sweep elements, the following equations are used for computing the each

component of the Jacobian matrix where the notations used in Eq. 7-8 are followed from

chapter 5.


dx
dr
dP(r, s,t) dy n dN, (r,s) M dN,2D (r, )
dr dr Y dr ~ dr
dz
dr

dx
ds
dP(r, s,t) dy a y dN,2D(r, ) M dN,2D(r, s)
= n u + mv, (7-8)
ds ds ds ds
dz
ds

dx
dt
dP(r, s, t) dy dR(t)
dt dt dt
dz
dt
If the point of interest is on the boundary of the element, the surface normal vector

is equal to the vector normal to the element face on which the point is located. The

position vector R(r, s, t) of an arbitrary point within the element is expressed as follows









n
x M, (r,s, t)


R(r,s,t)= y = yMl(r,s,t) (7-9)
1=1

nzZM l(r,s,t)
1=1
The six faces of a hexahedral element are represented as R(1, s, t), R(-1, s, t),

R(r,l,t), R(r,-1, t), R(r, s,1), R(r, s,-1) respectively. Since these are parametric

equations of surfaces, the normal vector for any point on these faces can be easy

computed as the cross product of the tangent vectors along the parametric directions. For

example, normal vector to the face R(1, s, t) is


aR(1, s, t) aR(1, s, t)
n x (7-10)
as at
Similarly, the normal vectors can be computed on the other five faces. Table 7-1

shows the normal vectors on all six faces of the element.






Table 7-1 Surface normal vectors on the boundary of the element
Element Face Normal Vector Element Face Normal Vector
R(1, s, t) cR(1, s, t) R(1, s, t) R(-1, s, t) OR(-1, s, t) X R(-1, s, t)
x x
as at at as
R(r,1, t) aR(r,l, t) R(r,lt) R(, t) t) 8R(r,-1, t) X R(r,-1,t)
at dr dr at
R(r, s,1) aR(r, s,1) XR(r, s,1) R(r,s,-1) 8R(r, s,-1) XR(r, s,-1)
x x
_r as _s Or


Surface normal is ambiguous for points along the edges and vertices of the implicit

solid element unless a face of the element is also specified whose normal is to be

computed at the point.









7.2 Set membership classification within an element

An important requirement for any unambiguous solid model representation is that

one should be able to classify a given point as being inside, on, or outside the solid.

Algorithms for such point set member classification are non-trivial for B-Reps. For

traditional implicit surfaces, one can use the value of the implicit surface function to

determine which side of the surface a point lies on. However, for primitives defined using

implicit solid elements, the density function is expressed as a function of the parametric

coordinates and therefore it is necessary to first map a point from the real space into the

parametric space to evaluate the density. The mapping between those two spaces is

typically non-linear. If the given point is X, then the following equations for the

parameters rp, s, and t, that corresponds to this point need to be solved.


Xp = XM,(rp,Sp,tp) (7-11)
If the mapping basis functions, M,(rp,sp,tp), are nonlinear three simultaneous non-

linear equations need to be solved. Since it is not always possible to obtain analytical

solution for such equations it is necessary to use numerical methods such as Newton-

Raphson method to find a solution. Clearly, these equations will be easier to solve if

simple basis functions for the mapping are used. For example, when a 9-node element is

used, the interpolation basis function is bicubic, but one could use a simpler bilinear (4-

node) basis functions for the mapping if the element edges are not distorted into curves

and mid-nodes are mid points of each edge. Moreover the mapping function can be

solved analytically to find the parametric coordinates from the given global coordinates

as shown in Chapter 4.









An analogous set of mapping equations for (x, y, z) can be derived for the 8-node

hexahedral element where trilinear basis functions are used. Even though for the general

case these equations are nonlinear, they reduce to linear simultaneous equations for

elements whose opposing faces are parallel. Therefore, if the shape of the element in the

real space is a parallelogram in 2D or a parallelepiped in 3D then the mapping is linear

and easy to perform in either direction.

However, in general, three nonlinear simultaneous equations need to be solved

numerically to compute the parametric coordinate. To avoid such numerical computation

3D problems are reduced to 2D problems that can be solved analytically. This is based on

the idea that the cross-sections of the implicit solid element along the sweep direction are

parallelograms and the mapping from the real space to the parametric coordinates of this

parallelogram is bilinear. All predefined 2D primitives are parallelograms so that the

mapping equations can be solved analytically.

Therefore the first step in mapping is to find the parametric value t for the given

point. When the sweep trajectory is a straight line, the parameter t* can be easily

computed by comparing the distance from the given point and the element front face and

the depth of the element. Similarly, when the sweep trajectory is an arc, the parameter t*

can be computed by comparing the angle between the given point and the element front

face and the total angle of revolution of the element.

In the case of sweep elements including sweep blend elements where the sweep

trajectories are arbitrary parametric curve equations, the nonlinearity in the mapping is

introduced by the sweep trajectory. The parameter t* can be computed as the optimal

solutions obtained by minimizing R(t) X that is by finding the closest point on the









trajectory R(t) from the given point Xp where the R(t) should satisfy the following

equation.


OR(t*)
R( (R(t*) X) = 0 (7-12)
at
Now, from the parameter t* obtained from the above procedure, a parallelogram

having the four vertices such as X(1, 1, t*), X(-1, 1, t*), X(-1, -1, t*) and X(1, -1, t*) can be

built. Thereafter analytical method can be used to find the parametric coordinate r and s.

By using this technique, set-membership classification is fast and efficient in this system.

After finding the parametric coordinates (r, s, t) from the given global coordinates

(x, y, z), the given point can be easily classified as inside, outside and on the solid with

respect to primitive by computing the shape density value at the point. Figure 7-1 shows

the procedure for set-membership classification with respect to an element as a flow

chart.


Figure 7-1 Flow chart for set-membership classification in an individual solid element









7.3 Constructive Solid Geometry using Implicit Solid Elements

As mentioned earlier, CSG representation is a well-established technique for

representing solid models and is used to construct solid models by Boolean combination

of primitives that are represented either as B-Rep or using implicit half spaces. The same

approach can be used with primitives represented by the implicit solid elements. An

advantage of using the implicit surfaces is that set membership classification is

computationally less expensive than with B-Rep. Furthermore, there is no need for

expensive boundary evaluation algorithm, which is needed in the hybrid approach to

automatically construct the B-Rep model of the solid represented by the CSG tree. This

step is necessary in the hybrid approach because the B-Rep model of the Boolean result is

needed for creating the graphical display and also for subsequent operations such as set

membership classification. However, if the primitives used in the CSG tree are

represented using implicit solid elements, it is possible to directly compute the solid's

properties, generate graphical display and perform set membership classification based

on the procedural description of the solid embodied in a CSG tree. The following figure

shows the CSG tree data structure used in this system.


O Boolean Node
O Primitive Node








(a) 2D CSG tree (b) 3D CSG tree
Figure 7-2 CSG tree data structure









The CSG tree shown in Figure 7-2 illustrates how a solid is represented in this

system. Every leaf node represents a primitive defined by implicit solid elements and the

branch node represents a Regularized Boolean type such as union, intersection and

subtraction (U /n / respectively). Before performing Boolean operations the two

participating solids need to be positioned correctly with respect to each other. This could

be achieved by translations and rotations specified by the user but for dimension driven

editing the required transformation must be computed based on specified positioning

constraints. In this implementation, a sequential constraint imposition method [Kumar,

2000] was used to compute the transformation matrix required to impose position

constraints between solids. Typically, position constraints are specified between vertices,

edges and faces of a solid if the solid is represented using a B-Rep model. Even though

such entities are not available in implicit solid models, the nodes, edges and faces of the

implicit element itself serve as useful datum that can be used for specifying constraints.

Figure 7-3 is a simple example where the 2D profile is obtained by the union of a

rectangle and a circle followed by the subtraction of an ellipse. Figure 7-4 shows a 3D

example.










i I LI


Figure 7-3 2D Boolean operation in CSG





































Figure 7-4 3D Constructive Solid Geometry Tree

For a primitive solid defined using a single element, the shape density value at a

point can be computed as used to classify the point as inside the solid, on the boundary of

the solid or outside the solid. When many primitives have been combined using Boolean

operations, a procedure for classifying points with respect to the Boolean results are

required. In order to apply the same procedure for the Boolean result the shape density at

every point within the solid including regions where multiple primitives may intersect

needs to be defined. This can be accomplished by using the following definition that is

commonly used in the literature on level set methods.


AUB = max(oA> B) Union

A B = min(A Bq) Intersection (7-13)
b g= min((^A, b) Intersection (7-13)









A-B =min(A ,-~B) Subtraction
In the above definitions, QACB is the density at any given point within the solid

defined as the union between two solids A and B whose shape density values are 0A and

B respectively. It can be easily verified that QA4B will be greater than the level set value

at every point within the solid defined by the union. It is equal to the level set value at the

boundary and less than equal to outside the solid. Similar definitions are given above for

the intersection and difference between solids.

Another way to define density at every point within a solid is by using R-functions

[Shapiro, 1998], [Shapiro, Tsukanov, 1999] which can be used to define the density for

the solid defined using Boolean operations. The sign of such a function depends only on

the sign of its arguments and not their magnitude. This property can be used to define

density values at points where two or more primitives intersect. However, to use R-

functions it is necessary that every point on the boundary of the solid have value equal to

the level set value bb. In implicit solid elements, the solids boundary can be the same as

element boundary where the density need not be the level set value. Therefore, to use R-

function method the definition of the implicit solid element will have to be modified

accordingly. In this implementation, R-functions to define density for Boolean solids

were not used.

As mentioned Chapter 2, when the given point is on both the base and the

dependent solids, it is difficult to determine its set membership with respect to the

regularized Boolean result by simply comparing its set membership with respect to base

and dependent solids. In order to apply the rules of regularized Boolean operations, it is

necessary to study the neighborhood of the given point to check whether it is on the









boundary or the interior of the solid. Alternately, one can compare the direction of

surface normal of the solids between which the regularized set operation is being applied

[Mantyla, 1988]. This approach can be applied recursively to perform set membership

classification for arbitrarily large CSG trees. By applying this procedure the density

functions only at the primitive level are used.

As mentioned in Chapter 2, the neighborhood model enables us to classify points

with respect to the regularized Boolean when multiple normal vectors are involved. From

a theoretical point of view, the neighborhood model at a given point is easy to understand

but it is very difficult to construct an algorithm based on this model since there are

infinite points in a neighborhood that needs to be checked. In this system, a graphical

method for constructing the neighborhood model is proposed and implemented. This

method is based on the assumption that the maximum number of normal vectors involved

at a given point is three in the level of solid elements and it is convex not concave.

Therefore triangles from the normal vectors that represent half spaces can be constructed.

The detailed description for constructing triangles from the surface normal vectors is

explained in the following.

In a solid element, triangles can be constructed from the surface normal vectors as

shown where a face of the solid has one normal vector, an edge of the solid has two

normal vectors and a vertex of the solid has three normal vectors respectively.


























Figure 7-5. surface normal vectors and corresponding triangles

When there is only one surface normal vector at a given point, a triangle normal to

this vector can be constructed by the following procedure.


Figure 7-6: Constructing a triangle from one normal vector

In the Figure 7-6, n is the unit surface normal vector ( n


1). a, b and c are unit


vectors perpendicular to the surface normal vector n. a can be chosen from any arbitrary

vector perpendicular to the surface normal n. Therefore, a should satisfy the following

condition.

an = and a= 1
Then point A can be computed as


A = P + sa


(7-14)









The scalar s is the distance from the given point P to the point A. Therefore it

defines the size of the triangle. Since the size of the triangle does not matter while

constructing neighborhood model, any value can be used for s. In this implementation 0.5

is used for s. After computing a, the vectors b can be computed by rotating a 120 degree

about axis n and similarly c can be computed by rotating b 120 degrees. Then points B

and C can be computed as.


B = P + sb
C = P + sc (7-15)
When the number of surface normal vectors is two, the corresponding two triangles

can be obtained by the following procedure. Let nl and n2 be the two unit surface normal

vectors at the point of interest.

n = 1 and n2 =1


ni






Edge b \ b d
B \ n2
BX\

D


Figure 7-7. Constructing two triangles from two normal vectors

a is computed as the cross product of ni and n2.


a = n x n2 (7-16)
b=-a

c and d can be computed by cross product.










c= n xa

d= n2 xb
To use c and d vectors for constructing triangles, their direction should be verified.

Since it is assumed that each solid primitive is the intersection of half spaces that are

normal to nl and n2 respectively, the following equations show its validity.


If n2 c < 0, c = c. if not c = -c.

If ni d I O, d d. if not d =-d.
After finding vectors a, b, c and d, the points composing vertices of the triangles

can be computed as


A=P+sa, B=P+sb, C= P+scand D=P+sd (7-17)
Finally, when the number of surface normal vectors is three, the corresponding

three triangles can be obtained by the following procedure. In Figure 7-8, ni, n2 and n3

are unit normal vectors at the point of interest.

ni =1, n2 =1 and n, =1


Edge E








EdgeEdge


Figure 7-8. Constructing three triangles from three normal vectors

a is computed as the cross product between ni and n3.


a = n1 xn, (7-18)

If n2 a < O, a = a. if not a = -a.

Similarly,









b= n2 xn1

If n3 *b < 0, b =b. if not b =-b.

c = n3 x n2

If ni c < 0, c = c. if not c = -c.
After finding vectors a, b, c and d, the points composing the vertices of the

triangles can be computed as

A=P+sa, B=P+sband C= P+sc (7-19)
After constructing the triangles for all the primitives whose boundaries pass

through the point of interest, the neighborhood at the point is constructed by performing

the Boolean operations between the triangles treating each triangle as a half space, which

divides the space into two regions inside and outside. At first, intersections between

triangles from the base and dependent solids are computed for each Boolean operation,

and then the triangles will be subdivided at the intersections. Thereafter set membership

classification of the triangles with respect to the half spaces (instead of the actual solid) is

performed to collect the valid triangles.

To classify the triangles, first the centroid Q of the triangle is computed and then

this point is classified with respect to Boolean result recursively starting from the bottom

of the CSG tree. It is assumed that the classification of the centroid is identical to the

classification of the triangle. In addition, subdivided triangles that do not contain the

point of interest are removed from the valid triangles because the neighborhood model

can be constructed without those triangles. If there are triangles at the final node, it means

that the set membership is on the solid. If there are no triangles, then it implies that the set

membership is in or out.

The above algorithm can be used for 2D or 3D CSG tree with the difference that

lines are used to represent half spaces in 2D. Figure 7-9 shows an example that shows






how to construct the neighborhood at a given point. As shown in Figure 7-9, the point of
interest from A and B has multiple normal vectors. Two triangles and three triangles are
derived from the solid A and B respectively. A union operation between those triangles is
performed to get the valid triangles at the union node where four triangles are obtained
after union. Therefore the point of interest can be classified as on the solid after union.




B AB
Figure 7-9. Neighborhood model using graphical method
Figure 7-10 shows examples of solids created by Boolean operations using this
software implementation




a' '