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GEOMETRY AND MAPPINGS OF SCREWS WITH APPLICATIONS TO THE HYBRID CONTROL OF ROBOTIC MANIPULATORS By HARVEY LIPKIN 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 1985 Copyright 1985 by Harvey Lipkin ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to his committee members, Professors Gary Matthew, George N. Sandor, Ralph Selfridge, John Staudhammer and Lou Torfason as well as to Professor Delbert Tesar, who have all helped make this work possible through their continued encouragement and support. Special thanks are given to the committee chairman, Professor Joseph Duffy, for his guidance and enthusiasm throughout all phases of this work. The author is indebted to Professor Kenneth H. Hunt for his liberal applications of geometric stimulation, both directly and through antipodal correspondence. Professor Eric J.F. Primrose is gratefully thanked for sharing his comments and geometrical insight on many topics. The author greatly appreciates the efforts of Ms. Denise Gater in typing this dissertation under such arduous chronological constraints. Further, the Roger Gater Courier Service has been most invaluable. Finally, the author has depended upon the moral support and motivation extended to him by his wife, Leslie, and son, Matthew. The financial support of the National Science Foundation (Grant No. MEA8324725) and of the Westinghouse iii Research and Development Center, Pittsburgh, PA, is gratefully acknowledged in making the completion of this work possible. TABLE OF CONTENTS Page ACKNOWLEDGMENTS ...................................... iii ABSTRACT ............................................. vi CHAPTER 1 INTRODUCTION.................................... 1 2 PROJECTIVE GEOMETRY .......................... 4 2.1 A Projective Formulation of Coordinates. 6 2.2 Homogeneous Coordinates ................. 19 2.3 Projective Transformations ............... 39 3 METRICAL GEOMETRY ............................ 57 3.1 Projective Metrics ........................ 62 3.2 Elliptic Geometry......................... 77 3.3 Euclidean Geometry............. .......... 88 3.4 Vector Formulations ..................... 110 4 INVERSIVE GEOMETRY......................... .. 132 4.1 The Elliptic Polarity of Screws.......... 135 4.2 Quaternion Mappings ..................... 146 4.3 Ball's Planar Representation of the TwoSystem.............................. 166 4.4 SelfPolar Screw Systems ................ 187 5 APPLICATIONS.................................. 205 5.1 Hybrid Control........................... 208 5.2 Noninvariant Methods .................... 225 5.3 Noninvariant Properties .................. 245 5.4 Invariant Formulations .................. 265 6 SUMMARY AND CONCLUDING REMARKS................ 285 BIBLIOGRAPHY ......................................... 290 BIOGRAPHICAL SKETCH .................................. 297 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 GEOMETRY AND MAPPINGS OF SCREWS WITH APPLICATIONS TO THE HYBRID CONTROL OF ROBOTIC MANIPULATORS By Harvey Lipkin August 1985 Chairman: Joseph Duffy Major Department: Mechanical Engineering The nature and invariant properties of the elliptic polarity of screws in Euclidean space is established using a new series of mappings, central to which is a quaternion representation. The mappings are used to generalize Ball's planar representation of the twosystem of screws which is shown to be a mapping on a complex plane where the elliptic polarity induces a conformal mapping. Screw theory is applied to the hybrid control of robotic manipulators where it is demonstrated that a current theory based on orthogonall" projection yields noninvariant results. New invariant methods of hybrid control are detailed by introducing invariant kinestatic filters. CHAPTER 1 INTRODUCTION Research may start from definite problems whose importance it recognizes and whose solution is sought more or less directly by all forces. But equally legitimate is the other method of re search which only selects the field of its acti vity and, contrary to the first method, freely reconnoitres in the search for problems which are capable of solution. Different individuals will hold different views as to the relative value of these two methods. If the first method leads to greater penetration it is also easily exposed to the danger of unproductivity. To the second method we owe the acquisition of large and new fields, in which the details of many things re main to be determined and explored by the first method. A Clebsch [1871, p. 6] The theory of screws was first established by Ball [1900] after a quarter century of development. Although most mathematicians of the late nineteenth century were acquainted with screw theory, very few actually pursued its development. Consequently, in the early twentieth century it virtually faded into obscurity. However, with a revived interest in spatial mechanisms in the 1960s and robotics in the 1970s, screw theory is being slowly res urrected in either the guise of dual vectors or Plucker coordinates. An excellent treatment of Plucker coordinates in application to mechanisms has been given by Woo and Freudenstein [1970]. Recent texts by Hunt [1978] and Bottema and Roth [1979] have investigated and applied screws in various kinematic areas. 1 However, the problem that was initially investigated for this dissertation, the orthogonalityy" of screws, was not treatable using the current literature in the field of kinematics. It was necessary to reconsider some of the geometrical work of the nineteenth century and try to put it in a modern perspective with improvements of notation, methods, etc. The beginning of this dissertation deals with a systematic development of the geometry of points, planes and lines to provide a broad basis for understanding the subtleties of screw theory in a unified framework. As such, the author has tried to integrate some of the pertaining accomplishments of the great late nineteenth century geometers: Ball, Cayley, Clifford, Grassmann, Klein and Plucker. When ever possible, original papers were referred to in order to understand the spirit and implications between the lines which are forever lost in most modern presentations. Ori ginal quotes have been included throughout to compensate somewhat for this lamentable circumstance. The geometrical development rightly begins with projec tive geometry, which is the most general, in Chapter 2. In order to commence on an analytical basis at the outset, coordinates are introduced in a metricfree manner for points. They are then generalized using determinant princi ples which elegantly display dualistic properties. Projec tive transformations leave incident relations invariant and are developed based on a tetrahedron principle. Metrical geometry is developed in Chapter 3 from pro jective geometry based on Cayley's conception of the Absolute. Elliptic geometry is detailed with emphasis on determining elliptic relations for screws that appear simi lar to projective ones. Here the elliptic polarity is first introduced. Elements of Euclidean geometry are then derived with what appears to be a new contribution in deducing Euclidean collineations based directly on the invariance of the Absolute. Vectors are introduced and are used to characterize screws in particular. Chapter 4 deals with the role of the elliptic polarity in Euclidean space. It uses the previous developments to explain the motivation for and the importance of this work. Unless specifically noted all developments in this chapter are claimed to be original. A series of new mappings of screws is introduced, central to which is a quaternion representation. Ball's planar representation of the two system of screws is generalized as a mapping on an inversive plane where the elliptic polarity induces a conformal mapping. In Chapter 5 screw theory is applied to the hybrid control of robotic manipulators. It is demonstrated that a current theory based on orthogonall" projection yields noninvariant results under Euclidean translations and changes in the unit of length. By way of introducing invariant kinestatic fil ters, new invariant methods of hybrid control are presented. CHAPTER 2 PROJECTIVE GEOMETRY Throughout the ages, from the ancient Egyptians and Euclid to Poncelet and Steiner, geometry has been based on the concept of measurement, which is defined in terms of the relation of congruence. It was von Staudt (17981867) who first saw the possibility of constructing a logical geometry without this concept. Since his time there has been an increasing tendency to focus attention on the much simpler relation of incidence, which is expressed by such phrases as "The point A lies on the line p" or "The line p passes through the point A." H.S.M. Coxeter [1942, p. 18] During the nineteenth century a great deal of investigation centered about geometrical relations which are independent of metrical concepts such as distance, angle, area, etc. Metricfree geometric relations form the subject of projective geometry of which the property of incidence is paramount, such as a point is incident to a plane or a line cuts a conic. For many years projective geometry was viewed as a relatively insignificant area within the domain of Euclidean geometry. The situation took a radical turn in 1859 when Cayley demonstrated that projective geometry was actually the most general and that Euclidean geometry was merely a specialization where a certain configuration was deemed fixed and used as a reference for measurement. Later, Klein demonstrated how nonEuclidean geometries could be included under Cayley's principle which is explained in Chapter 3. Thus, there arose a contradiction. Since coordinates were typically based upon metrical considerations, how could such coordinates be logically applied to projective relations since metrical geometry is subordinate? Klein supplied an answer to this by suggesting the use of von Staudt's projective constructions which are employed to define the algebra of points and is presented in a modified form in Section 2.1. First nonhomogeneous coordinates are introduced which are then used to define the homogeneous coordinates of points. In Section 2.2 determinant formulations are utilized to develop homogeneous coordinates in the plane for points and lines and in space for points, planes, lines and screws following extensionsal principles attributable to Grassmann. This enables a rather elegant and complete symmetry to be displayed amongst dual elements. Plucker's ray and axis line coordinates are detailed and their important idential relation is formulated which is subsequently used throughout the entire work. Projective transformations of space preserve incidence relations and are detailed in Section 2.3. An important tetrahedron relationship is introduced which unifies projective transformations for points, planes and lines or screws. Polarities are developed as correlations whose properties are symmetrical with their inverses and are then employed in Chapter 3. Section 2.1 A Projective Formulation of Coordinates In the first place, it is important to realize that when coordinates are used, in projective Geometry, they are not coordinates in the ordinary metrical sense, i.e. the numerical measures of certain spatial magnitudes. On the contrary, they are a set of numbers, arbitrarily but systematically assigned to different points, like the numbers of houses in a street, and serving only, from a philo sophical standpoint, as convenient designations for points which the investigation wishes to distinguish. But for the brevity of the alphabet, in fact, they might, as in Euclid, be replaced by letters. Bertrand Russell [1897, pp. 118119] Projective geometry deals with the incidence properties of elements and figures without employing any form of measure ment or metric. However, various metrical subgeometries may be derived from the more general projective geometry by re quiring that certain figures be designated as an absolute reference. Thus to employ metrical concepts, such as Cartesian coordinates, in the analytical development of projective geom etry is to incur a logical contradiction since metrical rela tions are subordinate. In this section, it is demonstrated that coordinates may be introduced into projective geometry in a metricfree manner. Coordinates are important in the analytical development of projective geometry since they are the tools for the derivation of compact expressions for complex relations and further, they provide insight which may not be evident from a synthetic de velopment. Coordinates also provide an essential vehicle for numerical calculations which may be used to verify and illus trate relations unambiguously. Since coordinates are routinely used in metrical geometries it is highly desirable to demon strate how they may be first utilized projectively and then specialized for metrical application. The introduction of coordinates in projective geometry presented here is not intended to be an exhaustive treatment but rather an illustrative one. The intent is to draw a sharp distinction between projective and metrical concepts at the outset and to provide a logical sequence of development from projective to metrical geometry. The analysis in this section is based primarily on the complete and systematic methods of Veblen and Young [1910, 1917] to whom frequent reference is made. It was however von Staudt [1857] who first demonstrated that analytic methods may be introduced into geometry on a strictly projective basis. For this purpose he invented the algebra of throws. The method was later simplified by Hessenberg [1905] and is presented here. First, the addition and multiplication of points on a line is defined using two special projective constructions. Then this algebra of points is demonstrated to be isomorphic to the field of real numbers and is extended to include the concept of infinity in a con sistent manner. A unique real number is associated with each point on the line with the exception of a single point which assumes a correspondence with infinity. The unique real number associated with each point is called the nonhomogeneous coordinate of the point on the line. The apparently exceptional role of the point associated with infinity is removed upon the introduction of homogeneous coordinates. In order to assign coordinates to points on a line U, it is first required to select three distinct points x0, x1 and x. which together are referred to as a scale. By the special nature of the constructions which define addition and multipli cation, the points of the scale are endowed with the properties of 0, 1 and . Figures 2.1.1 and 2.1.2 illustrate the construc tions for addition and multiplication which can also be used respectively to define subtraction and division. Using the operation of addition it is possible to label all the points corresponding to integers, e.g. x1 + x1 = x2, x1 + x2 = x3, etc. Next, using the integers with the operation of division it is possible to label all points corresponding to rational numbers. Thus it can be shown that the points corresponding to rational numbers together with the two con structions are isomorphic to the field of rational numbers which includes the properties of associativity, communitivity, distributivity and the existence of inverses. Next, it is necessary to show that for all points on the line there is a corresponding real number and that together with the constructions they are isomorphic to the field of real numbers. This transition is made from the rational points by invoking a Dedekind cut which is detailed in Coxeter [1942], Veblen and Young [1910]. For the notation used here, the nonhomogeneous coordinates of the points x0, x1, xa, are given by the subscripts 0, 1, a, where the point x is excluded. It is important to note that the constructions yield UIo U00 Uoo00 X0 Xa Xb Xa+b Xoo Figure 2.1.1 Addition of two points. A fixed line U0 through x0 meets the two distinct fixed lines U and U in points r and s respectively. The lines x r and x s meet U and U respec a b a tively at r' and s. The line r s meets U at x a+b which yields the sum of the two points, x + xb = x a+b. By reversing the latter steps, subtraction is analogously constructed, eg. xa = xa+b xb. Uo0 Xo X1 Xa Xb Xab Xoo Figure 2.1.2 Multiplication of two points. Through the points x0, x1, x. are drawn respectively the fixed lines U0, U U with U0, U1 meeting at r and U U. meeting at s The lines x r and Xb s meet U and U respectively at r and s. Line r s meets U at xab which yields the product of the two points, x xb = x ab. By reversing the latter steps, division is analogously defined, eg. xa = xab xb. a ab x + x = x (1) a 0 x x = x (x x ) (2) x x = x (xa x0 ) (3) which are consistent with the usual properties associated with infinity. Although the three distinct points comprising the scale are selected arbitrarily, the addition and multiplication constructions impart them with the special properties asso ciated with 0, 1 and w. However, from a projective standpoint, all points have identical properties. Most generally, the fundamental theoremof projective geometry, Schreier and Sperner [1935], states that in a space of dimension n, there is a unique projective transformation between a pair of n+2 elements providing that for each part of the pair no n+l ele ments belong to a space of dimension nl (.see Section 2.3 for n=3). For points on a line n=l and thus three distinct points are related by a projective transformation. Three distinct new points may be chosen as another scale and all other points relabelled in terms of it. By way of projective transforma tions, all scales and subsequently all coordinates are pro jectively equivalent. Nevertheless, when a particular scale is employed, the scale points will still have the special properites of 0, 1 and due to the definitions of the con structions. The approach taken here is to use the addition and multiplication constructions solely for the purpose of labelling the points. These constructions are not utilized further, such as to analytically express projective trans formations in terms of nonhomogeneous coordinates. Thus, the special properties of the scale points do not enter into any further analytical development. Projective homogeneous coordinates are introduced by associating a pair of numbers a0, al, which can be written T as the 2xl array [a0 a 1] with the nonhomogenous coordinate a of the point x such that a a1 a (4) a0 The homogeneous coordinates of xa are not unique since [Aa0 Aa1]T aT X0O also satisfies (4). The coordinates [0 X]T, X#0 are asso ciated with the point x Thus to every pair of homogeneous coordinates, with the exception of [0 0] there corresponds a unique point of the line and to every point of the line there corresponds a pair of coordinates, which to a scalar multiple, is unique. Projective homogeneous coordinates for points in a plane are developed from the homogeneous coordinates of points in three distinct lines. As illustrated in Fig. 2.1.3, three noncollinear points are selected as the vertices of a triangle of reference and are used to establish two scale points on each side, w0, w x0, x and y0, y.. A fourth point which does not lie on one of the edges is selected as the unit point. Lines through the unit point and each vertex establish on the sides opposite the unit points wI, x1 and yl, thus complet ing the three scales. Together, the three vertices and the unit [iI Y1 I, I /;* \ I I / \ s S "; .";'*.. I I Xl Xa [F iol Wi 10111T Woo Xoo Figure 2.1.3 Projective homogeneous coordinate for the plane. '~ point are said to form a reference frame and respectively their homogeneous coordinates are designated by [1 0 O]T [0 1 0]T [0 0 I]T and [1 1 I]T (5) Every point on a side of the triangle may now be assigned a triple of homogeneous coordinates [a a a a2]T. For example, on line U each of the scale points is characterized by a2=0. In terms of the homogeneous coordinates of U every point has the form [a0 al]T which is now denoted in the planar system by [a0 a1 0] Analogously, points on U and Ux have coordi w x nates of the form [0 a1 a2]T and [a0 0 a2]T. For any point not on a side of the triangle, all of its coordinates [a0 a1 a2] will be nonzero. Let its projections on U and U from the opposite vertices be the points x and x y a Ya with coordinates determined by their respective scales, [xa0 0 xa2 ]T and [ya0 al 0]T It is possible to select the coordinates of the point such that al Yal a2 x a2(6) 0 a0 a0 Xa0 that is in the ratios, a0:al:a2 = xaOY a0:xa0 al:Xa2 Ya0 The coordinates of xa and ya can thus be expressed respectively as [a0 0 a2T] and [a0 a 0]T For the system of coordinates to be consistent, it is necessary to demonstrate that the coordinates of wa the projection of point a on U ,are propor tional to [0 a a2]T This can be proved using an argument involving cross ratios which is detailed in Veblen and Young [1910, pp. 175176]. Therefore, to a nonzero scalar multiple, the projective homogeneous coordinates of a point on a plane are given by the triple [a0 a1 a2]T. It should be noted that the four points chosen for the reference frame, no three of which are collinear, are otherwise arbitrary. Thus any four such points may be chosen since by the fundamental theorem of projective geometry they are related by a unique projective transformation. The method of using coordinates on three lines to establish coordinates on the plane leads to a recursive algorithm that may be used for a space of any dimension and is briefly described here for three dimensions. Four noncoplanar points are chosen as the vertices of a tetrahedron of reference and a fifth point not incident with a face is selected as the unit point. As illustrated in Fig. 2.1.4, the four vertices and unit point are respectively assigned the coordinates, [1 0 0 O]T [0 1 0 0]T [0 0 1 O]T [0 0 0 1]T [o 0 0 ]T [ 0 0 0 T [0 1 0 01 Figure 2.1.4 Tetrahedron of reference for projective homogeneous coordinates in space. and [1 1 11T (8) Lines through the unit point and each vertex establish unit points on the opposite sides respectively, [0 1 1 1] [1 0 1 1]T [1 1 0 1]T [1 1 1 0] (9) and together with the vertices, they are used to establish a planar reference system on each face. Every point which is incident to a face of the tetrahedron may now be assigned a quadruple of homogeneous coordinates [a0 a a2 aa3] For example, on the face opposite the vertex [0 0 0 1]T, each of the points is characterized by a3=0. This face has precisely the same planar reference system (5) described previously and thus has the form [a0 a1 a2] which is now de noted in the spatial system by [a0 a1 a2 0] Analogously, points on the remaining faces are characterized respectively by a0=0, al=0, a2=0. For any point not incident to a face of the tetrahedron, all of its coordinates [a0 a1 a2 a3]T are nonzero. It is neces sary to show that the projections of the point onto the four faces can be assigned coordinates in the planar systems given respectively by [0 a1 a2 a3]T [a0 0 a2 a3 T [a0 a1 0 a3]T [a0 a1 a2 0]T (10) This has been proved by Veblen and Young [1910, pp. 194195]. It should be noted that the five points chosen for the reference frame, no four of which are coplanar, are otherwise arbitrary. Thus any such five points may be selected since by the fundamental theoremof projective geometry they are related by a unique projective transformation. In the preceding developments, only the projective homo geneous coordinates of points in spaces of one, two and three dimensions were considered. Onedimensional spaces are also described by lines in a plane through a point or planes through a line and an analogous procedure may be developed to assign coordinates. Further, for twodimensional spaces such as a lines in a plane or planes through a point and for three dimensional spaces where for example the assemblage of all planes forms a threespace, coordinates may be assigned. In the twodimensional space of a plane, it is possible to simultaneously assign point and line coordinates and in threespace it is possible to assign point and plane coordi nates without incurring any logical inconsistencies. In the next section such a methodology is presented and is then extended by determinant principles to assign coordinates to lines and screws in projective threespace. Section 2.2 Homogeneous Coordinates A point given by its coordinates and a point determined by its equation, or geometrically speaking by an infinite number of planes inter secting each other in that point, are quite different ideas, not to be confounded with one another. That is the case also with regard to a plane given by its coordinates and a plane repre sented by its equation, or considered as contain ing an infinite number of points. Hence is derived a double signification of a right line. It may be considered as the geometrical locus of points, or described by a point moving along it, and accordingly represented by two equations in x, y, z each representing a plane containing that line. But it may likewise be considered as the inter section of an infinite number of planes, or as enveloped by one of these planes, turning round it like an axis; accordingly it is represented by two equations in t, u, v, each representing an arbitrary point of the line. The passage from one of the two conceptions to the other is a dis continuous one. The geometrical constitution of space, hitherto referred either to points or to planes, may as well be referred to right lines. According to the double definition of such lines, there occurs to us a double construction of space. In the first construction we imagine infinite space to be transversed by lines themselves consisting of points. An infinite number of such lines pass in all directions through any given point; each of these lines may be regarded as described by a moving point. This constitution of space is admitted when, in optics, we consider luminous points as sending out in all directions luminous rays, or, in mechanics, forces acting on points in every direction. In the second construction infinite space is like wise regarded as transversed by right lines, but these lines are determined by means of planes passing through them. Every plane contains an infinite number of right lines having within it every position and direction, around each of which the plane may turn. We refer to this second con ception when, in optics, we regard, instead of rays, the corresponding fronts of waves and their conse cutive intersections, or when, in mechanics, accord ing to POINSOT'S ingenious philosophical views, we introduce into its fundamental principles "couples," as well entitled to occupy their place as ordinary forces. The instantaneous axes of rotation are right lines of the second description. J. Plucker [1865, pp. 725726] The dualistic properties of projective geometry may be elegantly expressed in an analytic manner by employing homogeneous coordinates and determinant principles. The main objective of this section is the development of projec tive homogeneous coordinates in space, particularly line coordinates and their extensions to screws. However, it is appropriate to commence with point and line coordinates in the plane since this provides a foundation for the applica tion of extensional principles to threedimensional space, Forder [1940]. A line passing through two points may be described as the join of the two points and dually, the intersection point of two lines may be described as the meet of the two lines. Essentially, the principle of duality in the plane is that incidence relations remain valid when the roles of points and lines are interchanged (along with an appropriate altera tion in terminology such as replacing join with meet). For example, the previous relations may be expressed as, a line (point) is the join (meet) of two points (lines). In the plane, line coordinates are developed by first demonstrating, as in Coxeter [1942, pp. 7880], that the condition for a point and a line to be incident may be expressed as the linear relation S0w0 + Slwl + S2w2 = 0 (1) This relation can also be expressed more compactly by Sw = 0 (2) for which S = [SO Sl S21T, w = [w0 w1 w2] (3) The 3x1 arrays S and w are respectively projective homogeneous line and point coordinates in the plane and are unique to a nonzero scalar factor, i.e. only the ratios of the coordinates are significant. Assuming in (2) that the coordinates S have constant values while the coordinates w are free to vary, then (2) represents the locus of points which are incident to the line S, or in other words, the equation of line S. Dually, if the coordinates w are assumed constant while the coordinates S are free to vary, then (2) represents the pencil of lines which are incident to the point w, or in other words, the equation of point w. Equation (2) may be applied to express line coordinates in terms of point coordinates and reciprocally. Let x and y be two given distinct points on S and let w be any variable point on the line. Then using (2) yields three equations [w x y] S = 0 (4) for which the vanishing of the determinant w x yj = 0 yields an equation in w w01x1 y21 + wllx2 Y01 + w2xo0 Yll = 0 (6) where lxi y.j = x. y. x. y. (7) is the ij minor in (5). Based on the expansion (6), it is useful to define the determinants of the nonsquare arrays [w] (or w) and [xy] as the 3x1 column arrays wI = [w0 w w2]T (8) xyl = [jx1 y21 Ix2 yol Ixo y ]T. (9) Thus equations (5) or (6) may be expressed simply as w Ixy = 0 or w Ixyl = 0 (10) Comparing (10) with (2) yields the coordinates of line S to a scalar multiple in terms of points x and y, S = jxyl. (11) Since x and y are assumed distinct, then from (5) it is readily deduced that any point on the line S may be expressed in the form w = ax + By (12) where scalars a and $ are both not simultaneously zero. Equation (12) is referred to as the freedom equation of the line and any point on it is given by the ratio a:3. Using (.12) it may be shown that the coordinates of S are, to a scalar factor, independent of which two distinct points are chosen. For example, replacing x and y with (a0x + 80y) and (alx + B1y) in (11) yields S = aB Ixyl (13) where the scalar aSj 0 since the points are assumed distinct. Because of the duality principle, the development of line coordinates in terms of point coordinates is completely analogous to the preceding analysis. Let T and U be two given distinct lines incident to w, then if the variable line S is also concurrent, [S T U]Tw = 0 (14) for which S T U = 0 (15) yields an equation in S, SOIT1 U21 + SlT 2 U0 + S2 ITO Ull = 0 (16) or equivalently, ,slI IT UI = 0 (17) Comparing (17) with (\2) yields the coordinates of point w to a scalar multiple in terms of lines T and U, w = IT U. (18) Analogous to (12), the freedom equation for any line incident to w is deduced from (15) S = aT + BU (19) where a and 6 are both not simultaneously zero. In Section 2.1 the vertices of the reference triangle were assigned point coordinates corresponding to the columns of 13 (where In is the nxn identity matrix) and the first column represented what is usually called the origin. By substituting pairs of vertices in (11) it is shown that the line coordinates of any side and the point coordinates of the opposite vertex are given by the same column of I3. The side opposite the origin is often called the line at infinity though no reference to distance is implied here. For the plane, point and line coordinates can be summarized using xi IT (20) x y IT U. (21) In (20), point and line coordinates are given respectively by the three lxl determinants of [x] and [T] selected in the order 0, 1, 2. In (21) line and point coordinates are given respectively by the three 2x2 determinants of [xy] and [TU] selected in the order 12, 20, 01. In higher order spaces many of the developments for the plane may be further generalized by application of extensional determinant principles which were first introduced by Grassmann, see Forder [1940], Klein [1908]. For threedimensional projective space, the point may be chosen as the fundamental element and then a line is the join of two points and a plane is the join of three points. Dually, the plane may be selected as the fundamental element and then a line is the meet of two planes and the point is the meet of three planes. In space, the point and plane are dual elements whereas the line is selfdual. The duality principle in space asserts that incidence relations that are valid for points, lines and planes remain valid when their roles are interchanged with planes, lines and points along with an appropriate alteration in terminology. The condition for a point and plane to be incident may be expressed as the linear relation SOw0 + S1w1 + S2w2 + S3w3 = 0 (22) or equivalently Sw = 0 (23) for which now S = [S0SIS2S3 T w = [w0wlW2w3] (24) The 4x4 arrays S and w are respectively projective homogeneous plane and point coordinates and are unique to a nonzero scalar factor since only the ratios of the coordinates are significant. Equation (23) may represent either the locus of points incident to plane S or dually, the bundle of planes incident to point w, the difference being respectively whether S or w is assumed fixed while the other is free to vary. Plane coordinates may be developed from point coordinates by requiring that one variable and three given noncollinear points be incident to a plane [w x y z]Ts = 0 (25) for which w x y zI = 0 (26) yields an equation in the variable point w w0 xly2z3l + wllx2yOz3 + w2x0oylz3l w3jx1l0Z21 = 0 (27) and Ixiyjzkl is the ijk minor in (26). Alternately, (27) may be expressed more compactly using nonsquare determinants wIT Ixyz = 0 or wT xyzl = 0 (28) where IwI = [w0 w1 w w3]T (29) Ixyz = [Ixl 2z31x2yOz3l1xoylz3 xlyOz2]T (30) Comparing (30) with (23) yields the coordinates of plane S to a scalar factor in terms of the points x, y and z, S = Ix y zI. (31) Since x, y and z are noncollinear, then from (26) it is deduced that any point on the plane S may be expressed in the form, w = ax + By + yz (32) where the scalars a, 6 and y are all not simultaneously zero. Equation (32) is the freedom equation of the plane and any point on it is specified by the ratios a: ( :y. Substituting for x, y and z in (31), any other three noncollinear points on the plane S, a.x + $.y + y.z, i=l,2,3 yields S = ae y Ixyz (33) where the scalar jacSyj 0 since the points are noncollinear. Briefly, by the principle of duality, the development of plane coordinates from point coordinates is entirely analogous to the preceding development. Let one variable plane and three given planes, which themselves do not meet in a line, all be concurrent at a point [S T U V]w = 0 (34) for which IS T U V = 0 (35) yields an equation in the variable plane S, S01T1U2V31 + SI1T2UoV31 + S21TOU1V31 + S3IT1U0V21 = 0 (36) or equivalently S I T ITU V = 0. (37) Comparing (37) with (26) yields the coordinates of point w to a scalar factor in terms of planes T, U and V, w = IT U V1. (38) Further, the freedom equation of the planes through w is deduced from (35), S = aT + BU + yV (39) for a, 8 and y not all simultaneously zero. In Section 2.1 the vertices of the reference tetrahedron were assigned coordinates corresponding to the columns of 14* By substituting triples of vertices in (31) it may be shown that the plane coordinates of any face and the point coordinates of the opposite vertex are given by the same column of 14. For instance, the first column corresponds to the vertex at the origin and to the face that is often referred to as the plane at infinity. In space, point and plane coordinates may be summarized and line coordinates introduced using the nonsquare determinants, x IT (40) Ix y IT U (41) Ix y z T U V (42) In (40) point and plane coordinates are given respectively by the four lxl determinants of [x] and [T] selected in the order 0, 1, 2, 3. For (42), plane and point coordinates are given respectively by the four 3x3 determinants of [x y z] and [T U V] selected in the order 123, 203, 013, 102. Plucker's ray and axis line coordinates, Plucker [1865, 1866], are defined by (41) respectively as the six 2x2 determi nants of [x y] and [T U] selected in the order 01, 02, 03, 23, 31, 12. Ray line coordinates p represent the join of two points x, y and axis coordinates P represent the meet of two planes and P= P01P02P03P23P31P12 'T j = Ixijl (43) = 3T T.U. (44) P = [P01P02P03P23P31P12 Pij = (44) or more briefly, p = x y P = IT U (45) Referring to Fig. 2.2.1, the relationship between ray and axis coordinates is derived from the incidence relations of two points with two planes, Tx = 0 (46) TTy = 0 (47) Ux = 0 (48) U y = 0. (49) Forming in turn, T (48) U *C(46), T (49) U (47), x (47) y *(46) and x *(49) y (48) yields respectively, P,p Figure 2.2.1 A line is formed as either the join of two points or the meet of two planes. [P ]x = 0 (50) [P*]y = 0 (51) [p*]T = 0 (52) [p*]U = 0 (53) where the rank two skewsymmetric arrays are given by 01 02 03 P P P P P P P30 P31 P32  P01 P02 P03 [p*]= 10 12 13 (55) 20 P21 P23 P30 P31 P32 _ From (50), (51) and (54), the rows (or columns) of [P ] are four planes through the line, each incident with a vertex of the reference tetrahedron. From (52), (53) and (55), the rows (or columns) of [p ] are four points on the line, each incident with a face of the reference tetrahedron. Forming either (50). y (51)* xT or (52). UT (53). TT yields the matrix equation [P*][p*] = 0 . (56) 04 04 04 04 o r1 04 r r4 0 C) . + + + + Q4 a4 04 04 i 0 0 H (i CD DI fri + 0 Q4 N N C4N N CD "m ( Y 04 04 04 04 rq H o H 04 CN 04 04 04 04 + + + + H 0 H 0 H4 0 HI 0 o C 0 C( ) + C04 04 C4 04 OY m mm m N 04 04 0 0 a4 04 04 04 + 0 0 0 o  r i 0 o (Y) II II II 0 ~N 0N O0 II II II r N 0 C n o ii o i o 11 a4 04104 0 4 a I 04 I II II I4 rL4 a4 1 a4 C 4 33 In Table 2.2.1, (56) has been expanded into components to yield (57). Then the offdiagonal terms are used to form the twelve equations in (58) written as a 4 x 4 array. Comparing in turn five equations in (58) given by the positions 30, 10, 13, 03, 01 yields P P P P P P 23 P 31 P 12 P 01 P 02 = P03 .... p (5 9 ) 01 02 P03 23 P31 12 where i is a nonzero scalar. It should be noted that it is not necessary to explicitly include p since it is included implicitly by only ascribing significance to the ratios of homogeneous coordinates. Thus setting i=l and arranging (59) in matrix form yields P = Ap (60) p = AP (61) where 13 A = (62) 13 A = A, AA = 16. (63) Since in (60) and (61), p and P are derived from dual elements, the induced linear transformation A is a correlation which is signified by tilda. The existence of this correla tion is due to the fact that in threedimensional space lines are selfdual elements and thus A represents the identical correlation of lines. The matrix A represents a very simple method of transforming between ray and axis coordinates, the first and last three components are merely exchanged. Since a double application of the exchange yields back the initial values, then as expressed by (63), A is clearly an involution matrix. In ray coordinates, the condition for two lines to be incident may be obtained by letting p be the join of w, x and q be the join of yz. When the lines intersect then all four points are coplanar and w x y zI = 0 repeated, (26) which is expanded by the first two columns to yield p A q = 0 or lw xl A ly z = 0. (64) Alternatively, in axis coordinates let P and Q be the same two lines where P is the meet of X, T and Q is the meet of U, V. When the lines intersect then all four planes are concurrent at a point and the expansion of S T U VI = 0 repeated, (34) yields PT A Q = 0 or IS TIT AJU VI = 0. (65) Using (60), (61) with (64), (65) gives alternative expressions for intersection, p TQ = 0 or Lw xlT LU V, = 0 (66) PTq = 0 or Is TIT x y = 0. (67) Because dual coordinates are used in (66), (67) their form is analogous to (23) which expresses the incidence of a point and a plane. Since only the ratios of homogeneous coordinates are significant, the six coordinates of a line represent five parameters. However, only four independent parameters are required to specify a line in space and therefore six coordi nates must be related by a single equation. In terms of ray and axis coordinates, this relation is obtained by expanding the singular determinants Ixy xy = 0 TU TU = 0 (68) to yield T T p A p = 0 PA P = 0 (69) or equivalently p P = P Tp = 0. (70) It is interesting to examine the line coordinates cor responding to the tetrahedron of reference. Referring to Fig. 2.2.2, the vertices along with their opposing faces are both labelled 0, 1, 2, 3 and their coordinates correspond in order to the columns of 1I. Forming the ray coordinates of the six joins of vertex pairs and the six meets of face 3 23 03 31 2 a) 12 001 1 \ 2 02  b)3 03 23 3 Figure 2.2.2 The tetrahedron of reference labelled with a) point coordinates and ray coordinates, b) plane coordinates and axis coordinates. pairs both in the order 01, 02, 03, 23, 31, 12, then the result ing coordinates of both sets correspond in order to the columns of 16. Opposite edges of the tetrahedron (i.e. nonintersecting), one expressed in ray coordinates and the other in axis coordi nates, correspond to the same column of 16' Unlike point and plane coordinates, there are in general no freedom equations for lines using line coordinates correspond ing to (32) and (39). (There are however freedom equations for lines in terms of either two points or two planes.) For example, let q,r be two lines and 11,A be two scalars and consider p = pq + Xr (71) For p to be a line it must satisfy the quadratic form of (69). However, T T p A p = 2 p q A r (72) which only vanishes when q and r intersect. In general, a linear combination of lines in either ray or axis coordinates is defined here as a screw, which includes lines as special cases. Since the homogeneous coordinates of a screw need not satisfy (69), it follows that there are 5 screws in space. When two screws satisfy any of the equivalent relations T = 0 PT Q (73) p Aq=0 ,P AQ= (73) T P Q = 0 T q = 0 , Qq = 0 (74) the screws are said to be reciprocal, a property which is analogous to incidence. For a general linear combination of n screws (or lines), 1 < n < 6, the freedom equations p = q + r (75) P = XIQ + R where X1 n are all not simultaneously zero, are said to describe an nsystem of screws, Ball [1900], Hunt [1978]. Screws may also be viewed abstractly as points in a five dimensional space as suggested by F. Klein (see Jessop [1903] ). Lines in this space lie on a surface which is given by (69). Dually, hyperplanes in this space correspond to fivesystems of screws, which also may be described by six homogeneous coordinates. By generalizing the development of coordinates presented here, systems of coordinates may be systematically derived for nsystems, 1 < n < 5. Detailed references on the generation of extensional systems of coordinates based on determinant properties are given by Forder [1940], Sommerville [1929], Hodge and Pedoe [1947, 1952], but will not be considered here further. Section 2.3 Projective Transformations I shall enunciate two general principles which I have habitually emphasized and have put into the foreground in these fundamental geometric discussions. Although in this generality they sound at first somewhat obscure, they will, with concrete illustrations, soon become clear. One of them is that the geometric properties of any figures must be expressible in formulas which are not changed when one changes the coordinate system, i.e. when one subjects all the points of the figure simultaneously to one of our transformations; and, conversely, any formula which, in this sense, is invariant under the group of these coordinate transformations must represent a geometric property. As simplest examples, which all of you know, let me remind you of the expression for the distance or for the angle, in the figure of two points or of two lines. We shall have to do repeatedly with these and with many other similar formulas in the following pages. For the sake of clearness, I shall give a trivial example of noninvariant formulas: The equation y = 0, for the figure consisting of the point (x,y) of the plane, says that this point lies on the x axis, which is, after all, a thoroughly unessential fact, foreign to the nature of the figure, useful only in serving to describe it. Likewise, every noninvariant equation represents some relation of the figure to external, arbitrarily added, things, in particu lar to the coordinate system, but it does not re present any geometric property of the figure. The second principle has to do with a system of analytic magnitudes which are formed from the coordinates of points 1, 2, such as X, Y, and N, for example. If this system has the property of transforming into itself, in a definite way, under a transformation of coordinates, i.e., if the system of magnitudes formed from the new coordinates of the points 1, 2, expresses itself in terms exclusively of these magnitudes formed in the same way from the old coordinates (the coordinates themselves not appearing explicitly), then we say that the system defines a new geometric configuration, i.e., one which is independent of the coordinate system. In fact, we shall classify all analytic expressions according to their behavior under coordinate transformation, and we shall define as geometrically equivalent two series of expressions which transform in the same way. Felix Klein [1908, pp. 2526] This section examines linear projective transformations of homogeneous coordinates. Klein [1872] has enunciated a definition of geometry which, except for minor extensions, is still very applicable today. Essentially, Klein stated that a geometry is defined as the properties of a space which remain invariant under all transformations of space (or the coordinate system) by a group of transformations. For projective geometry, the group of transformations is characterized by those which preserve relations of incidence. Commencing with the group of projective point collineations, the corresponding induced collineations for planes, lines and screws are developed with respect to an elegant tetrahedronal principle employing determinant relations. Using a simple device, many of the results for collineations are extended to the nongroup of correlations. An analysis of projective trans formations not only identifies important invariant relations but also forms a foundation for developing metrical geometries in Chapter 3. A collineation is a onetoone linear transformation in which each element of space is mapped into a corresponding element of the same type (e.g. point to point) whereas a cor relation differs in that each element is mapped into a corres ponding dual element (e.g. point to plane). A projective transformation is uniquely determined by five pairs of corresponding points in space provided that no four of the five points in either pair are coplanar. For the collineation, 41 it = Kx (1) where p is included explicitly as a factor of proportionality and where the 4x4 matrix K is given by K = [A B C D]T (2) Since only the ratios of homogeneous coordinates are signifi cant, the four equations in (1) can be reduced to three ratios of equations by, for example, dividing the last three equations by the first equation and thus the explicit factor p is eliminated. Multiplying out the ratios and expressing them in matrix form yields T T tx tx A T T t2x t0x B = [0] (3) tx T t0xT C 3 0 D where t0 t3 are the coordinates of t and where the 3x16 matrix multiplies the 16x1 column array containing the unknown coefficients. Substitution for t and x by five pairs of corresponding points yields 15 homogeneous equations which are sufficient to solve for 15 ratios involving the elements of K. Thus the projective collineation is uniquely determined to a scalar factor and K is nonsingular since the mapping is onetoone. In (1) the factor p was explicitly included to facilitate in the solution for K. However, it is convenient to absorb the factor by substituting u=i which is permissible provided it is understood that only the ratios are significant, t = Kx. A projective collineation of points also induces a projective collineation of planes which may be determined using incidence properties. Let x be incident to plane T, Tx = 0 (5 The induced transformation k maps T into another plane X X = kT (6 such that incidence is preserved, XTt = 0. (7 Substituting (4) C6) in (7), TT(kTK)x = 0 (8 and comparing with [5) yields k = KT (9 to an arbitrary nonzero scalar multiple. Matrix k can be calculated by replacing each element of K with its cofactor (signed minor), and dividing by the scalar IKI = IA B C DI (although this last step is not essential), k= [lB C DI IC A DI IA B D IB A C] T/IA B C DI (10) or more simply as k = [a b c d] (11) The four 4x3 determinants in (10) have been formed from ABCD in the order 123, 203, 013, 102 which is also the same order used in expanding each of these nonsquare determinants into components, see (2.2.42). There is a useful geometric interpretation for K and k. Let ABCD represent the coordinates of four planes whose equa tions can be written as [A B C D]Tw = 0. (12) Since K is nonsingular then (12) has no solution other than w=0 which does not represent a point in homogeneous coordinates. Thus the four planes do not have a common point and they there fore form a tetrahedron. In (10), each row is the meet of three planes and is thus a vertex of the same tetrahedron. The vertices abcd are respectively opposite the faces ABCD since (2), (9) and (11) yield the incidence relations [A B C D]T [a b c d] = 14 (13) Additionally, K can be expressed in terms of k and by analogy with (10), K = [lb c dl Ic a di ja b dl lb a cl]T/ a b c d (14) The collineation of points not only induces a collineation of planes, but also induces a collineation of lines. Let x,y and t,u be a pair of corresponding points t=Kx u=Ky (15) Forming the join of each pair, then the lines are expressed in ray coordinates by p = Ixyl q = Itul (16) and the line p is transformed into the line q. Substituting (15) into (16) yields q = jKx Kyl and then substituting (2) in (17) gives the nonsquare determinant T ATx BT x CTx DTx ATy BTy T C y D y (17) (18) The first coordinate of q is given by q01 = ATxB y A yB x = AT[xyT yxT]B = A [p*]B (19) where [p ] is the skewsymmetric metrix given in (2.2.55) where elements are p.ij = x.y.. Expanding the bilinear ij 1 J expression in (19) yields after some manipulations, q01 = AB Tp. (20) The remaining components of q are determined by analogy with (20) which yields q = [IABI ACl ADI ICDI IDBI IBCl]Tp (21) and which is more concisely expressed by q = Kp. (22) The six 4x2 determinants in (21) have been formed from the planes ABCD in the order 01, 02, 03, 23, 31, 12 which is also the same order used in expanding each of the nonsquare determinants into components, see (2.2.41). Analogously, the induced collineation for axis coordinates can be developed from a pair of corresponding planes, X = kT Y = kU (23) Forming the meet of each pair in axis coordinates, L = ITUI M = IXYl (24) then the collineation transforms line L into line M and using (23) in (24) yields M = kT kUl. (25) Expanding the terms in (.25) and using (11) gives a result which is analogous to (21), M = [jab lacl lad Icdl Idb l Ibc ]T L (25) or more concisely, M = k L. (26) Since a collineation preserves incidence properties, then if p and L are incident then so are q and M, LTp = 0 (27) MTq = 0 (28) Substitution of (22), (26) in (.28) yields T ^T L (k K)p = 0 (29) which leads to a result analogous to (9), ^ ^T k = K (30) upon comparing (27) and (29) for general intersecting lines L,p. Although (30) is correct to a scalar multiple, it can be shown that it is actually an algebraic identity when the plane collineation is given by (2) and (14) and the point collineation is given by (10) and (11). Figure 2.3.1 is used to illustrate the geometric inter pretations of the induced ray and axis line collineations. From (21), the collineation of ray coordinates is given by K whose rows are the axis coordinates for the six edges of the illustrated tetrahedron. From (25), the collineation of b) Figure 2.3.1 IauI C C I Ibcl lab I b I AC \ I AD I The tetrahedron associated with a collineation labelled with a) point coordinates and ray coordinates, b) plane coordinates and axis coordinates. A axis coordinates is given by k whose rows are the ray coordi nates for the six edges of the tetrahedron. Thus the same tetrahedron is intimately related to the four collineations K, k, K, k. This tetrahedron relationship leads to two important algebraic identities for the induced line collineations, ^T ~ ^ K AK =K ^T ~ ^ k A k = A B C DI A ja b c dIA. . I3 Briefly, the identities (31) (32) are the conditions for the rows of K and k to be edges of the tetrahedron in Fig. 2.3.1. These identities are demonstrated by first substituting (30) in to (31) (32) to yield a b c dIA = k A kT A B C DjA = K A K and where additionally a b c dl A B C DI = 1. (33) (34) (35) Relations (33), (34) are easily proved by substituting in the values of k and K in (21) and (25) and by noting that, where (31) (32) (2.2.62) IabI A Icd = a b c d lAB IA CD = JA B C DI (.36) (37) since the left sides are merely the Laplacian expansions of the right sides by the first two columns. Any other product such as T a ab A ad = Ja b a d = 0 (38) clearly vanishes since two columns are identical. Using incidence properties, (31) and (32) are now veri fied to a scalar multiple. Since p, q, L, M are lines then they satisfy the identical relations given by (2.2.69), T ~ p A p = 0 T q q A q = 0 T ~ L A L = 0 MT AM =0. Substituting (22), (26) into(.40) yields T T = 0 p (K A K)p = 0 T ^T ~ L (k A k)L = 0. Comparing (39) and (41) for independent p and L yields (31), (32) to a nonzero scalar multiple, ^T A ^= A K A K = pA ^T A ^ ~ k A k = p]A. Equation (42) (or (31), (32)) is important since it yields the conditions which are necessary for a 6x6 matrix to represent an induced projective collineation of lines. (39) (40) (41) (42) A 6x6 matrix contains 36 elements and since only the ratios are significant there are 35 independent parameters. In (42), either of the equations represents only 21 different scalar equations since the matrix equations are symmetric. Because these equations are not all homogeneous, the scalar multiplier p is eliminated by considering the 20 ratios of equations (in a manner analogous to (3)). Thus the 35 parameters are related by 20 constraint equations to yield 3520=15 independent parameters to describe an induced projective collineation of lines. This is in agreement with the projective point and plane collineations whose associated 4x4 matrices yield 15 independent ratios. A linear combination of lines, as previously defined by p = X q + Ar (2.2.75) P = XQ +. AR (2.2.76) is in general a screw. The induced projective collineation of screws in space is identical to that for lines by way of linearity, Kp = XAKq + A Kr kP = 1 kQ + A kR. (43) If a 6x6 matrix does not satisfy (42) then it is not an induced projective collineation of lines. It is possible for this to occur in two ways. Firstly, if the rows of the matrix all represent lines then they cannot form a tetrahedron. Secondly, if any row is a screw then the matrix is not an induced collineation. General transformations of this type have been investigated by Ball [1900], which he called homographic transformations. It should be noted that in the literature the terms homographic transformations and projective transformations are often used synonymously but are used distinctively here. Effectively, Ball treated screws as points in a projective fivespace and therefore a homographic collineation is the most general onetoone linear transformation of these points. It is not difficult to verify that nonsingular pro jective point and plane collineations each form a group of transformations under the operation of composition or matrix multiplication. Nonsingular induced projective line collineations also form a group of transformations and it is useful to demonstrate the property of closure. Using ray coordinate transformations let K, J be two nonsingular AA collineations and it is necessary to show that KJ is also such a collineation, i.e. it satisfies (42), ^^ ~ ^^ ^T ^T ~ ^ ^ (KJ) A (KJ) = J (K A K)J ^T ^ = pj A J = iXA (44) where yX is a nonzero scalar multiplier. Since induced collineations constitute a group, then it follows from (42) that the bilinear forms, T ( T p A q ,P A Q (45) and in particular the quadratic forms T ( T p A p AP (46) are invariant expressions with respect to induced collinea tions. From linear principles, this is true whether p,q (P,Q) are lines or screws. The forms (45) are often referred to as the mutual moment of two lines or screws although no metrical connotation is implied here. When screws are considered as points in a fivespace then the lines of threespace are represented as points on the quadric surface T pT ( p A p = 0 P AP= 0 (47) which is sometimes called a Grassmannian, Hodge and Pedoe [1952]. In the group of homographic transformations of screws (i.e. nonsingular 6x6 matrices), induced projective line transformations constitute a subgroup which leaves the quadric (47) invariant (transforms into itself). Projective transformations are classified as either collineations or correlations. Correlations are linear onetoone transformations which map each element into a dual element. Since the product of two correlations is a collineation, it follows that correlations do not possess the property of closure and thus do not form a group. However, collectively collineations and correlations form the group of projective transformations. In the development of correlations it is useful to first define two distinct spaces where elements in one are denoted by a prime. Consider the incidence of a point x on a plane T, T x = 0 and the correlation, X = Ax t = XT. (47) (48) Since projective transformations must preserve incidence relations then, t X = T (X A)x = 0 (49) and by comparing (47) and (49) then to a scalar multiple, S= AT X = A A (50) The inverse transformations between the two spaces are given by inverting (48), T = X t' , x =A X (51) or equivalently using (50), T = A t T = A t x = T 1 x = X. (52) Figure 2.3.2 illustrates the mapping between the two spaces described by (48) and (52). It is noted that the two point Figure 2.3.2 The transformation of two distinct spaces under a correlation. to plane transformations (solid lines) are transposes of each other as are the two plane to point transformations (dotted lines). Correlations of a single space onto itself may be deduced by allowing the two distinct spaces to coincide. For the correlation to be welldefined it is necessary that TA = A = XT (53) for which either i = 1 and A,X and skewsymmetric or p = 1 and A,X are symmetric. Skewsymmetric correlations are called null polarities and since the matrix is of an even order it is generally nonsingular. Null polarities have many interesting properties, especially in relation to the linear complex, Jessop [19031, Busemann and Kelly [1953]. Symmetric correlations are referred to as polarities and are used to establish metrics in Section 3.1. Generally, the only correlations that are employed here subsequently are polarities. The development of polarities (or more generally correla tions) is facilitated by introducing the polarity I , I I = I (54) n n n A where I is the nxn identity collineation. A polarity may n be expressed as a product of In and a symmetrical collinea tion k,K, A = I4k = kI4 (54) X = I4K = KI4. (55) In this manner the results obtained for collineations may be applied directly to polarities. The induced polarity of lines corresponding to k and K is respectively I6k = kI6 (56) bA A I6K = KI (57) Analogous to (42), induced polarities have the tetrahedron property. For example, using (56) and (42) ~ ^ T ^ ^T T (I6k) A (I6k) = k (I6 A I6)k ^ T ^ = k Ak = pA. (58) In Chapter 3 it is shown that polarities may be employed in the development of Euclidean and nonEuclidean geometries using Cayley's conception of the Absolute. CHAPTER 3 METRICAL GEOMETRY Space is another framework which we impose on the world. Whence are the first principles of geometry derived? Are they imposed on us by logic? Lobatschewsky, by inventing nonEuclidean geometries, has shown that this is not the case. Is space revealed to us by our senses? No; for the space revealed to us by our senses is absolutely different from the space of geometry. Is geometry derived from experience? Careful discussion will give the answerno! We therefore conclude that the principles of geometry are only conventions; but these conventions are not arbitrary, and if transported into another world (which I shall call the nonEuclidean world, and which I shall endeavor to describe), we shall find ourselves compelled to adopt more of them. In mechanics we shall be led to analogous conclusions, and we shall see that the principles of this science, although more directly based on experience, still share the conventional character of the geometrical postulates. So far, nominalism triumphs; but we now come to the physical sciences, properly so called, and here the scene changes. We meet with hypotheses of another kind, and we full grasp how fruitful they are. No doubt at the outset theories seem unsound, and the history of science shows us how ephemeral they are; but they do not entirely perish, and of each of them some traces still remain. It is these traces which we must try to discover, because in them and in them alone is the true reality. H. Poincare [1905, pp. xxvxxvi] Cayley has shown how metrical concepts may be introduced into geometry on a purely projective basis. That is, a figure such as a quadric surface is designated as a fixed reference, the Absolute, and metrical properties, are those properties of figures which take on significance in relation to the Absolute. This is the starting point for the systematic development of metrical geometries in Section 3.1. Metrical relations are developed using projective coordinates and hence the seemingly selfcontradictory name of projective metrics. This section simultaneously treats the common properties of hyperbolic, Euclidean and elliptic geometries in a general manner using the notion of an Absolute polarity as an invariant connection of dual elements in space. Using a definition by Clifford, an analytical generalization for determining the pitches and axes of screws is given which apparently may be also found in Buchheim [1884b]. Metrical collineations are defined as those which leave the Absolute invariant and form a subgroup of projective collineations. Norms are then introduced as functions of the Absolute and enables the development of metrical coordinates where the components themselves are significant not just their ratios. Elements of projective space are then assigned a norm of unity although, in the general case, this leads to two sets of metrical coordinates for an element which differ in sign. Elements with a nonunity norm are defined as new types of space elements which have an associated weight or magnitude. Since the properties of the various metrical geometries vary considerably, Section 3.2 deals exclusively with elliptic geometry. First, the elliptic polarity is introduced which has a close connection with interpreting the coordinates of a space element in terms of dual coordinates and forms a basis for some of the developments in Chapter 4. It is shown that properties which are often erroneously associated with ndimensional "Euclidean" spaces, such as orthogonalityy," are actually properties of elliptic space when homogeneous coordinates are utilized. This is particularly important with respect to some later developments dealing with the orthogonalityy" of screws which actually signifies that two screws are elliptic conjugates. Specializing a previous formulation, it is shown that screws in elliptic space have two axes which are elliptic polars and two respective pitches that are reciprocal. These results agree with ones given by Clifford [1873] along with the notion that screws of pitch 1 have properties of free vectors. Other investigations dealing with screws in elliptic space are given by Buchheim [1884a, 1884b], Cox [1882], Heath [1885] and Ball [1900]. In collineation form, the elliptic polarity of lines and screws is similar in appearance to the important identical relation between ray coordinates and axis coordinates which is a symmetrical correlation. Table 3.2.1 summarizes a number of relations in elliptic geometry which appear very similar to expressions in projective geometry. This exemplifies why it is necessary to have an unambiguous notation to delineate collineations from correlations and ray coordinates from axis coordinates. A systematic development commencing with projective geometry makes it possible to delineate the distinct but similar expressions. Euclidean geometry is distinguished by the fact that the Absolute polarity is singular and consequently many relations must be approached as limiting cases. As shown in Section 3.3, the singularity introduces an asymmetric character to dual expressions which does not exist for projective or elliptic geometry. The general formulation in Section 3.1 is specialized for an interesting development of the pitch and axis of a screw which is expressible as a linear combination of a unique line and its polar. It is also shown that what is often referred to as the "dual" operator w, where w2 = 0, is merely the Euclidean polarity expressed in form amenable to biquaternions. Based on the Euclidean Absolute, norms are defined which are then used to introduce Euclidean metrical coordinates. Elements of projective space are assigned norms of unity and points are given a unique set of coordinates, unlike planes, lines and screws which have two sets of coordinates that differ in sign. New space elements are defined which have the property of weight or magnitude since their norms are nonunity. It is typical to study a geometry in terms of examining its group of transformations. However, the group of transformations in Euclidean space always appears to be an entity given a priori from which subsequent geometric properties are then derived. Here, beginning with the general group of projective collineations and Cayley's Absolute, it is shown how the corresponding group of Euclidean transformations may be deduced as those which leave the Euclidean Absolute invariant. Although the procedure is not complex, it appears to have been previously overlooked. In Section 3.4 polar and axial vectors in Euclidean space are introduced by way of Klein's second principle given at the beginning of Section 2.3. Then using Hamilton's vectors, a polar vector is defined as the difference between two points. By introducing vectors and making the point a more fundamental element in Euclidean space than the plane, the ambiguity of signs for the metrical coordinates of planes, lines and screws is examined. The ambiguity is only resolvable for new space elements that are then introduced namely, planesects, geometric couples, line vectors and screw vectors which are all distinguished by a magnitude and an unambiguous associated direction. The ray coordinates and axis coordinates of line vectors and screw vectors are then expressed in a formulation which is typical of modern presentations, especially the ones using dual vectors such as in Brand [1947]. Finally, in application to the area of mechanics, twists and wrenches are introduced along with the formulation of virtual work. When a body is in static equilibrium under impressed wrenches its virtual work vanishes, a property which is shown to be analogous to the projective property of incidence. Section 3.1 Projective Metrics I remark in conclusion, that, in my own point of view, the more systematic course in the present introductory memoir on the geometrical part of the subject of quantics, would have been to ignore altogether the notions of distance and metrical geometry; for the theory in effect is, that the metrical properties of a figure are not the pro perties of the figure considered per se apart from everything else, but its properties when considered in connexion with another figure, viz the conic termed the Absolute. The original figure might comprise a conic; for instance, we might consider the properties of the figure formed by two or more conics, and we are then in the region of pure descriptive geometry: we pass out of it into metrical geometry by fixing upon a conic of the figure as a standard of reference and calling it the Absolute. Metrical geometry is thus a part of descriptive geometry, and descriptive geometry is all geometry, and reciprocally; and if this be admitted, there is no ground for the consideration, in an intro ductory memoir, of the special subject of metrical geometry; but as the notions of dis tance and of metrical geometry could not, without explanation, be thus ignored, it was necessary to refer to them in order to show that they are thus included in descriptive geometry. Arthur Cayley [1859, pp. 592] In the preceding chapter, it has been demonstrated that homogeneous coordinates may be introduced into geometry without recourse to a form of measure, or in other words, a metric. Homogeneous coordinates are wellsuited for examin ing incidence relations which comprises the domain of projec tive geometry, or "descriptive geometry" as Cayley referred to it. Metrical geometries such as elliptic, Euclidean and hyperbolic may be developed from projective geometry by establishing one or more figures as a fixed reference, which Cayley called the Absolute. In threedimensional space, various metrical or socalled CayleyKlein geometries may be developed by defining the Absolute to be a point locus together with a plane envelope of a quadric surface. Projective homogeneous coordinates may be adapted for metrical geometries and in doing so they may also be en dowed with the additional property of magnitude, which is a function of the Absolute. It is most interesting to note, that prior to the landmark paper "A Sixth Memoir on Quantics," Cayley [1859], projective geometry was considered merely to be a somewhat poorer subject in what was then the allpervasive geometry of Euclid. Initially, not even Cayley recognized the scope of his dictum, "descriptive geometry is all geometry," since he had only considered the geometries known to him at that time, Euclidean and spherical, the former of which he pre sented for only one and two dimensions. It was left to F. Klein [1871, 1873], some twelve years hence, to demon strate that the elliptic geometry of Riemann and the hyper bolic geometry of Lobatchewsky and Boylai, the socalled nonEuclidean geometries, may be developed by selecting the Absolute to be respectively an imaginary or real figure. In this section, metrical geometries are developed in a general format which may then be specialized to yield hyperbolic, Euclidean and elliptic geometries, the latter two of which are investigated in the succeeding sections. Instead of commencing the development here with an Absolute quadric, it is preferred to first establish an Absolute polarity and its adjoint, Coxeter [1965]. 64 A correlation is a linear transformation which maps each element of space into a dual element. A polarity is a symmetric correlation which can be represented by a symmetri cal matrix and may be utilized to establish an invariant connection of space between dual elements. This is essen tially equivalent to the CayleyKlein development since there is generally a onetoone relationship between polari ties and quadric surfaces. It should be particularly noted that since metrical geometries are specializations within projective geometry, that metrical geometries must preserve projective properties, in particular, relations of incidence. The diagonal polarity T and its adjoint H where 1 . 1 TT = (1) S 1 1 11 n = (2) are used to establish elliptic, Euclidean and hyperbolic geometries for = 1,0,l respectively. Since the polarities become singular in the Euclidean case, the appropriate de velopment considers the limiting case 0. For the polar relations, X = 7x (3) x = rX (4) the plane X is said to be the polar of point x and the point x is said to be the pole of plane X. Two points x,y (two planes X,Y) are said to be conjugate when each is in cident with the other's polar (pole) and T x T y = 0 (5) XT Y = 0. (6) Sometimes conjugate points and conjugate planes are referred to respectively as Horthogonal or 'rorthogonal. The Absolute quadricc) is defined as the locus of selfconjugate points and the envelope of selfconjugate planes, T ~ x H x = 0 (7) XT i X = 0. (8) The polarity of points and planes induces a correspond ing polarity of lines. In Fig. 3.1.1, the join of points x,y is the line p and the meet of their polar planes X Y defines the polar line P Alternatively, the meet of planes T,U I U also defines the same line P and the join of their poles t ,u also defines the same polar line p Therefore P and p , the axis and ray coordinates of the polar line, are given respectively by the nonsquare determinants (see Section 2.2), P = IHx Hy P,p Figure 3.1.1 A pair of polar lines. P = IX TTY (10) Substituting, (1), C2) in (9), (10) and expanding yields the induced polar relations P = Fp (11) p = YP (12) where 3 F = (13) 3 3 Y = (14) I3  In (13) the common factor c has been removed which is necessary for the Euclidean case where E 0. Alternatively, (13) may be derived by substituting the relations between ray and axis coordinates (2.2.48), (2.2.49) in (14) to yield r = A Y A. (15) Conversely, (15) can be rearranged as ~ ~T ~ ~ Y = A r A. (16) Since r, y are symmetrical and are adjoints, then to a scalar multiple E, F Y = 16 (17) which may be used with (15), (16) to yield the tetrahedron relationships (see Section 2.3), A F = A (18) Y A y = A. (19) Two lines p,q (P,Q) are said to be conjugate when each is incident with the other's polar line and T 0 p F q =0 (20) T ~ P Y Q = 0. (21) Alternatively, lines which are conjugate are sometimes referred to as Forthogonal or yorthogonal. Lines which are self conjugate are incident with their own polars and their assem blage forms the tangent lines to the Absolute. In line coordinates, the Absolute is given by the quadratic complex, Jessop [1903], T ~ p F p = 0 (22) T ( P Y P = 0. (23) As previously defined, a linear combination of lines is in general a screw . A r P = 1 Q + . n 1 n R. (2.2.65) n p = \ q + . 69 Analogous to (11), (12), the polar of a screw is given by P = q + X r , p =X1 y Q + n y R (24) and two screws which satisfy a bilinear relation of the form (20) or (21) are also said to be conjugate. In describing screws in elliptic space, Clifford [1873, pp. 193] asserted that a screw can be expressed uniquely as the sum of a line and its polar line and that this polar pair represents the axes of the screw. Here this result is generalized and, using ray coordinates, a screw p is expressed as a linear combination of a unique line q and its polar q' i i p = Xq + X q (25) where it is necessary to determine the scalars X,X and the i I pair of polar lines q,q The line q may be expressed in ray coordinates by q = A Q = A F q (26) which is substituted in (.25) to yield p = (XI6 + X A F)q. (27) Provided that the matrix in (27) is nonsingular, then using (13) q is easily solved for and q = (XI X A F)p. (28) (X2_X,2E) 6 Since q is a line, it must satisfy the identical relation q A q = 0 (29) which is used to eliminate q. Substituting (28) in (29) yields, after some rearranging, 1 T ~ 2 ~ [(PT A p)X 2(pT p)XX + e(p Tp)A ]= 0 (30) (\2 X 2 E)2 For e = 1,0,1, (30) may be solved for the ratio X:X which is then substituted back in (28) to determine q and subse quently q' from (26). By extension of Clifford's definition, the ratios X :X resulting from (30) are called the pitches of the screw with respect to its axes. In the following sections the solution is detailed for both elliptic and Euclidean geometries. Metrical collineations are defined as those which leave the form of the Absolute invariant. For points and planes, the projective collineations of space (see Section 2.3) are given by y = Kx Y = kX. (31) In the image space, the Absolute must be expressible in the same form as (7), (8) and T y H y = 0 (32) Y iT Y = 0. (33) Substituting (31) in (32) and (33) yields T T~ x (K 1 K)x = 0 (34) X (k T k)X = 0. (35) Comparing (34), (35) with (7), (8) for general x,X gives T K TK = p H (36) T~ k 7rk = TT (37) where p is a nonzero scalar multiple. Equations (36), (37) express the required conditions for a nonsingular collineation to leave the Absolute invariant. Collineations which satisfy (36) or (37) are referred to as metrical collineations and they form a subgroup within the general group of projective collineations. A projective collineation is expressed using 16 elements and since only the ratios are significant, it is thus determined by 15 independent parameters. However, a metrical collineation must also satisfy (36) or (37), and by symmetry either matrix equation represents a set of 10 nonhomogeneous scalar equations from which p may be eliminated by considering the 9 ratios of equations. Thus the 15 parameters of a general projective collineation are related by 9 constraint equations to yield 159=6 independent para meters for the specification of a metrical collineation. Briefly, for the induced collineations of lines or screws, let K and k be respectively the ray and axis transformations, q = Kp (38) Q = kP. (39) The equations of the Absolute are T~ T ^T ^4 q Fq = p (K r K)p = 0 (40) T T QT Q = P(k y k)P = 0 (41) and comparing with (22), (23) yields K r K = iF (42) k y k = vy (43) which are the conditions for an induced collineation to be a metrical collineation. The establishment of an Absolute enables the introduction of a projective norm which is useful in the development of metrical coordinates. For points and planes respectively, the norms are given by the scalar functions jx 1 = (x T H x)2 (44) X 1 = (X 7T X)2 (45) and for both lines and screws, the norms in terms of ray and axis coordinates are Hp 11= CpT r p) (46) 73 IlP I = (pT Y P) (47) Normed elements are defined as elements whose norms are unity. Once an Absolute is established, it is possible to re move the restriction from homogeneous coordinates that only the ratios are significant. This transition from projective homogeneous coordinates to metrical homogeneous coordinates is initiated by first normalizing all elements. For example, the projective homogeneous coordinates of a point x are normalized by x_ x (48) iIx c(x x) Replacing x with a scalar multiple Xx, which also designates the same point in projective coordinates, yields Xx Xx = + x (49) 11 x 1 1 I  X ii  x 1i x where the norm of the scalar is XI II = (X ) (50) Therefore in general, the sets of projective homogeneous coordinates which correspond to a single element and differ by a nonzero scalar factor, are transformed by the normaliza tion into two sets of homogeneous metrical coordinates which designate the same element yet differ in sign. By including further constraints, it is possible to resolve this ambiguity of signs completely for hyperbolic geometry, only partially for Euclidean geometry and not at all for elliptic geometry, Busemann and Kelly [1953]. For the case of elliptic geometry, this situation has interesting consequences which are detailed in Section 3.2 together with the Euclidean case in Section 3.3. It is convenient to sometimes refer to elements of projective space, i.e. points, planes, lines and screws, as unit or unweighted elements since in terms of metrical coordinates they have unity norms. Since all the elements of projective space have a representation in terms of normalized metrical coordinates, it is reasonable to assign a new meaning to metrical coordinates with a nonunity norm. Such coordinates are said to represent weighted elements, Forder [1940], which are purely metrical in nature, i.e. they have no projective representation and are thus a new species of space element. Every weighted element may be represented as a scalar multiple of an unweighted element, e.g. for a weighted point x = Ix (51) 1Ix I1 where the weight is simply the norm. Oftentimes the terms weight and magnitude are used synonymously. A common physi cal example of a weighted element is a point mass. Metrical collineations have already been defined as a subgroup of collineations which preserve the form of the Absolute and have been formulated using projective coordi nates in (36), (37), (42), (43). However, these formula tions only define collineations uniquely to a scalar multi ple and are thus not suitable when it is desired to employ metrical coordinates with associated weights. For this purpose, it is required that a collineation also preserves the norms of weighted elements. Letting K and k be respec tively point and plane collineations, then it is required that the relations T K I K = T (52) k T k = 7 (53) be satisfied identically, not just to a scalar factor. For distinctiveness, a metrical collineation which preserves the norm may be referred to as a unit or un weighted collineation or if the context is clear, simply as a collineation. Weighted collineations do not preserve the norm and are not considered here further. Since a unit collineation may not be multiplied by an arbitrary nonzero scalar factor, it represents 16 parameters. Further, because (52) and (53) are symmetrical, either relation represents 10 independent scalar equations and thus, a metrical collineation is specified by 1610=6 independent parameters, a result that agrees with the previous one employing projective coordinates. For the induced unit collineations of lines and screws, let K and k be respectively the ray and axis trans formations then it is required that the relations KT ~ ^ K F K = (54) (55) be satisfied identically, not just to a scalar factor. It may be demonstrated that the collineations induced from unit collineations are also unit collineations. First, taking the determinants of (52) (54) yields k y k = y IKI2 = 1 "K2 = 1 which are the collineations the identical expressed as ^T ~ ^ K AK = k Ak =k Ik2 = 1 k = 1 (56) (57) necessary and sufficient conditions for the to be unweighted. Using (2.3.2) and (2.3.4), relations (2.3.31) and (2.3.32) may be IKA k A and taking the determinants of these equations yields K 2 6 K kl2 6 k = *k (.58) (59) (60) (61) Substituting (56) in (60), (61) yields the desired relations (57). The properties of hyperbolic, Euclidean and elliptic space vary considerably and the preceding general analysis only uncovers relations which are common to all. In the following two sections, specific properties are detailed which, in particular, distinguish elliptic and Euclidean geometries. Section 3.2 Elliptic Geometry Consider any vertical line, and a series of hori zontal planes cutting it at right angles. In ordinary or Euclidean geometry these planes intersect on the horizon, which is a straight line infinitely distant. In the geometry of a space of constant positive curvature, or elliptic geometry, the horizon is at a certain finite distance in all directions from the vertical line with which we started; it belongs to that particular line, which is called its polar, and is not the same for all vertical lines. Although it appears to be a great circle when viewed from the neighborhood of its polar, yet if we were to go to it and examine it we should find it straight. Points of it which are in opposite directions from a point on the polar are really identical; and every straight line in this space resembles a circle in being of finite length, so that if we travel far enough along it we shall arrive at our starting point. Every straight line has a polar line, which is the in tersection of all planes at right angles to it. Let us take a very small circle on a sphere, and suppose it to expand, keeping always the same centre. At the beginning the circle will be concave inside and convex outside; but when the expansion has gone on far enough it will become a great circle of the sphere, which is of the same shape on both sides, or is straight so far as the surface of the sphere is concerned. So if in Euclidian space we take a sphere and suppose it to expand, keeping always the same centre, it will continue to be concave inside and convex outside so long as it is finite; but when the radius has become infinite, the inside in one direction is the same as the outside in the opposite direction, opposite points being identical; thus the sphere is of the same shape on both sides, or is a plane, viz., the plane at infinity. In elliptic space, just as in geometry on the surface of a sphere, this takes place for a finite length of the radius, not for an infinite length; for every point there is a sphere having its centre at that point, which is also a plane. Or, which is the same thing, every point has a polar plane which is the locus of all points situate at a certain distance from it; this distance is called a quadrant. So also every plane has a certain point, called its pole, which is distant a quadrant from every point in the plane. All lines and planes perpendicular to the plane pass through its pole, and conversely. The polar lines of all lines in the plane pass through its pole, and so do the polar planes of all points in the plane. When two lines are polars of one another, every point of one is distant a quadrant from every point of the other; hence the polar planes of all points on one pass through the other. Every line which is at right angles to one meets the other, and conversely. W.K. Clifford [1876, pp. 390391] In relation to hyperbolic and Euclidean geometries, elliptic geometry has the simplest and most symmetrical properties. The results of the previous section are recounted for elliptic geometry by setting =l. The elliptic polarity for points and planes is nI = 14 (1) T1 = I4 (.2) and the polar relations are given by X = HIx = I4x (3) x = T1 X = I X. 1 Two points or planes which are elliptic conjugates satisfy respectively x 1 y = x 14 y = 0 (5) xT iT Y = XT I4 Y = 0. (6) The locus of selfconjugate points and the envelope of self conjugate planes define the elliptic Absolute T T x H x = x I x = 0 (7) X T X = X I X = 0. (8) 1 4 For ray and axis coordinates, the induced elliptic polarity is given by r1 = 16 (9) yl= 16 (10) and the polar relations are P = F1 p = 16 p (11) p = Y P = 6 P. (12) Two lines which are elliptic conjugates satisfy PT 1 q pT i6 q = 0 (13) T T ( P Y Q = 6 Q = 0 (14) and the line equations of the elliptic Absolute are T T ( p 1 p = p 6 p = 0 (15) T T P Yl P = P i6 P = 0. (16) By linearity, the elliptic polarity of screws is given by P = 11 q + n r r = A I q + n 16 r (17) P= X1 Y1 Q + X n Y R = A 16 Q + n 16 R (18) and two screws which satisfy (13) or (14) are also said to be elliptic conjugates. For points, planes, lines and screws the elliptic polarity is specified by a correlation which is the identity matrix, either 14 in (1), (2) or 16 in (9), (10). Thus, the elliptic polar of an element is determined by interpret ing its coordinates in terms of dual coordinates, (3), (4), (11), (12), (17), (18). Two elements which are elliptic conjugates (5), (6) (13), (14), are also said to be either ellipticorthogonal or more specifically prefixed by the polarity such as for points, IHIorthogonal or I 4orthogonal. From the equations of the Absolute, (7), (8), (15), (16) it is clear that the points, planes and lines which form the 81 Absolute are not real and thus it is often referred to as an imaginary or virtual quadric surface. The elements forming the Absolute are said to be elliptic selfconjugate or alternatively, ellipticisotropic. For lines and screws, it is very important to carefully distinguish projective relations from the metrical elliptic relations which are similar in appearance due to the ele mentary form of the elliptic polarity. The projective correlation between ray coordinates and axis coordinates P = Ap (2.2.60) p = AP (2.2.61) may be substituted in (11), (12) to express the elliptic polarity as a collineation P = 16 p = 16 A P = AP (19) p = 16 P = I6 A p = Ap (20) where the product of correlations I6A yields the collineation A = (21) 3 Further, substituting (2.2.60) and (2.2.61) in (13), (14) yields alternative formulations for ellipticorthogonality, T ~ T T~ ~ ^ p 16 q A 6 q = P A q = 0 (22) P I6 Q = p A I6 Q = p A Q = 0. (23) Table 3.2.1 summarizes the projective relations along side the elliptic relations which are very similar in form. As throughout, a tilda is used to signify that a transforma tion is a correlation, e.g. I 6, whereas a caret denotes a collineation, e.g. 16. In the table, each expression con tains either a correlation or a collineation and its correspondent contains the other. Further, for each pair of expressions, one of the two screws (or lines) which correspond are in dual coordinates while the other two are in the same coordinates. Without a clear notational distinc tion between correlations and collineations and between ray and axis coordinates these relations are easily confused and misinterpreted. It was first discovered by Clifford [1873], that in elliptic space a screw p can in general be expressed as a linear combination of a unique line q and its polar q , p = Xq + X q (3.1.25) both of which are called the axes of the screw. The ratios i i X :X and X:X are respectively the pitches of the screw with respect to q and q Setting E=l in the general formu lations (3.1.27), (3.1.28) and (.3.1.30) yields p = (XI6 + X A)q (24) Table 3.2.1 Projective and metrical relations that are similar in appearance Projective 1. Transformations of ray and axis coordinates P = Ap p = AP p = I6 p P = I P 6 Metrical 1. Transformations of the elliptic polarity i  p = Ap P = AP P = I6 p p = I6 P 2. Reciprocity 2. Ellipticorthogonality T p A q = 0 P A Q = 0 T ^ p I6 Q = 0 T 6 = 0 P1 I6 q = 0 T ^ TA T ^ p A Q= 0 T ~ P I6 q = 0 q = 6 [ X A]p (A2 p) 2)2 I p 1 [(pTAp)X 2 2(T I6 p)XX + (pTAp)X1 ]= 0 ( X2 2) 2 i2 Assuming first that XA \ be expressed by 2 1 A 2bAA + X = 0 T p 0 then (26) may and p A p 3 0 then (26) may (27) where T ~ p 16 p b = . p Ap The solutions to (27) are (28) 2 12 X = b + (b 1) 2 or equivalently 1 = i, 1 where 2 1) u = b + (b 1)2 2 1 , b (b 1)2 Substituting in turn the roots (30) back in (25) yields the corresponding pair of axes (32) 2) 6 (1 ) 6 (25) (26) (29) (30) (31)  JILIP A q/P 7 2 E16 PAJp (33) and since (A)q = ql/ (.34) the two axes are elliptic polars where the subscripts denote the associated pitches. When p A p = 0, p is a line and assuming that T p I6 p i 0 in (26), then either X=0 or \ =0 and thus from I I (3.1.25), respectively either p=X q or p=Xq. 2 !2 The exceptional case occurs when X =X and thus the pitch of the screw is either +1 or 1. The form of this screw p may be determined by first taking the polar of (3.1.25), I I I p = Xq + A q (35) and then subtracting and adding in turn (3.1.25) and (35) yields (p p) = (XX ) (q q) (36) (p +p) = (+X ) (q 4q) (37) When the pitch is +1 then (.36) yields p=p or p=Ap (38) and thus the first and last three components of p are the same. When the pitch is 1 then (37) yields p = p or p = Ap and the first and last three components are the same except that they have opposite signs. As a consequence of these special forms, screws of pitch +1 may be expressed in an infinite number of ways as the sum of a line and its polar. The axes of such screws are thus indeterminant and for this reason Clifford referred to them as vectors, a right vector when the pitch is +1 and a left vector when the pitch is 1. Clifford also made the observation that a screw p, of any pitch, may be expressed as the sum of a right vector and left vector uniquely, I I p = (P+ + PP (39) 2 2 Vectors are closely related to right and left parallel lines in elliptic space which are also called paratactics, Sommerville [1929], or Clifford parallels, but are not developed here. The transition from projective coordinates to elliptic metrical coordinates is made by first introducing elliptic norms. By setting =l in (3.1.44) (3.1.47), the elliptic norms for points, planes and lines or screws become x 1 = (xT 4 x) (40) 1X I, = (XT I X) (41) p 1 = (pT I6 p) (42) 11P I = (PT I6 P) (43) As discussed in Section 3.1, by normalizing projective coordi nates, elements in projective space become unit or unweighted elements in terms of metrical coordinates. In general, an unweighted element has two representations in metrical coordinates which differ only in sign. For elliptic space, there is no additional criteria that can be imposed which enables a definitive selection of sign without ambiguity. The resolution to this predicament is merely to associate the pair of coordinates with each element, e.g. x and x for a unit point. For the elliptic plane this has an interesting consequence since it enables modeling in Euclidean space by a unit sphere. That is, a point on the elliptic plane is modeled either by a pair antipodal points on a unit sphere or equivalently by a diametrical line through these points. A line on the elliptic plane is represented by a great circle. The analogy in space is somewhat more involved, however Clifford's description given at the beginning of this section is highly suggestive. Elliptic collineations are defined as the subgroup of projective collineations which leave the elliptic Absolute invariant. In terms of metrical coordinates, the unweighted collineations for points, planes and lines or screws satisfy the relations given by setting E=l in (3.1.52) (3.1.55), K I1 K = I4 (44) kT 14 k = I4 (45) K I K = I (46) 6 6 k I k = I (47) 6 6 Analogous to the convention for elements, a collineation has two distinct representations which differ by sign, e.g. K and K for point collineations, see Busemann and Kelly [1953]. From the above relations it is easy to de duce that the determinant of an elliptic collineation is +1 or 1. However, both representations of a collineation have the same determinant since the matrices are of an even order, e.g. KI = (1)41K = K. Further, it is readily deduced from (44) (47) that the inverse of an elliptic 1 T collineation is equal to its transpose, e.g. K = K In the literature such matrices are usually referred to as orthogonal or orthonormal. However, for distinctiveness elliptic collineations are referred to here as elliptic orthogonal or prefixed with the polarity such as 14 orthogonal. Section 3.3 Euclidean Geometry We shall find throughout this period, that almost every important proposition, though misleading in its obvious interpretation, has nevertheless, when rightly interpreted, a wide philosophical bearing. So it is with the work of Cayley, the pioneer of the projective method. The projective formula for angles, in Euclidean Geometry, was first obtained by Laguerre, in 1853. This formula had, however, a perfectly Euclidean character, and it was left for Cayley to generalize it so as to include both angles and distances in Euclidean and non Euclidean systems alike. Cayley was, to the last, a staunch supporter of Euclidean space, though he believed that non Euclidean Geometries could be applied, within Euclidean space, by a change in the definition of distance. He has thus, in spite of his Euclidean orthodoxy, provided the believers in the possibility of nonEuclidean spaces with one of their most powerful weapons. In his "Sixth Memoir upon Quantics" (1859), he set himself the task of "establishing the notion of distance upon purely descriptive principles." He showed that, with the ordinary notion of distance, it can be rendered projective by reference to the circular points and the line at infinity, and that the same is true of angles. Not content with this, he suggested a new definition of distance, as the inverse sine or cosine of a certain function of the coordinates; with this definition, the properties usually known as metrical become projective properties, having reference to a certain conic, called by Cayley the Absolute. (The circular points are, analytically, a degenerate conic, so that ordinary Geometry forms a particular case of the above.) He proves that, when the Absolute is an imaginary conic, the Geometry so obtained for two dimensions is spherical Geometry. The correspon dence with Lobatchewsky, in the case where the Absolute is real, is not worked out; indeed there is, throughout, no evidence of acquaintance with nonEuclidean systems. The importance of the memoir, to Cayley, lies entirely in its proof that metrical is only a branch of descriptive Geometry. The connection of Cayley's Theory of Distance with Metageometry was first pointed out by Klein. Klein showed in detail that, if the Absolute be real, we get Lobatchewsky's (hyperbolic) system; if it be imaginary, we get either spherical Geometry or a new system, analogous to that of Helmholtz, called by Klein elliptic; if the Absolute be an imaginary pointpair, we get parabolic Geometry, and if, in particular, the pointpair be the circular points, we get ordinary Euclid. . Since these systems are all obtained from a Euclidean plane, by a mere alteration in the definition of distance, Cayley and Klein tend to regard the whole question as one, not of the nature of space, but of the definition of distance. Since this definition, on their view, is perfectly arbitrary, the philosophical problem vanishesEuclidean space is left in undisputed possession, and the only problem remaining is one of convention and mathematical convenience. Bertrand Russell [1897, pp. 2930] In contrast to elliptic space, for which the elemen tary form of the polarity yields many symmetrical relations amongst dual elements, the singular nature of the Euclidean Absolute yields highly unsymmetrical dualistic properties. As a consequence, various general formulas which may be specialized for elliptic and hyperbolic geometry, do not appear applicable to Euclidean geometry unless they are treated as limiting cases where c0. One example is the formula for the distance between two points which yields an indeterminant result unless an infinitesmal device is utilized as in Klein [1908]. However, for most of the developments here it is possible to set E=0 in the general formulations of Section 3.1. The Euclidean polarity for points and planes is given by I . S = 1 (1) T0 (2) (As throughout, zero elements in an array are often denoted by periods and I1 is used here in place of 1 for greater symmetry.) The polar relations are X = 10x = x = x = 7TX = X = where the notation X and x is introduced, x = [xI x2 xT x = [xi x2 x 3 . From (3), for a point x not on the plane at infinity, x # [0 xI x2 x3 T, its polar plane X is in fact the plane at infinity, X = [x 0 0 0]T For a point x on the plane at infinity, x = [0 x1 x2 x3] its polar plane X is in determinate since all of its components vanish, X = [0 0 0 0]T. From (4), for a plane X which itself is not the plane at infinity, X 3 [X0 0 0 0]T then its pole x is a point on the plane at infinity, x = [0 X1 X2 X3] For X itself being the plane at infinity, X = [x 0 0 0] its pole x is indeterminate x = [0 0 0 0]T Two points or planes which are Euclidean conjugates satisfy respectively, T ~ T 1 x 10 y = x y= x0 0 = 0 (7) X T y=T 0 T Y = XT Y = 0. (8) 3 In (7) two points are Euclidean conjugates if either lies on the plane at infinity. In (8), two planes which are Euclidean conjugates are said to be Euclideanorthogonal and thus meet at right angles. The angles of selfconjugate points and the envelope of selfconjugate planes define the Euclidean Absolute T ~ T 1 2 x I0 x = x x = x0 = 0 (9) T ~ T 2 2 2 X TX 0 X = X = XX = X + X + X (10) I3 Equation (9) represents the plane at infinity taken twice which is a rank one quadric locus. Equation (10) represents what is referred to as the imaginary spherical circle since it is the intersection of every sphere with the plane at infinity and further, because it is a rank three quadric envelope it represents a conic, see Klein [1908], Sommerville [1934]. Thus unlike elliptic geometry, the locus and enve lope of the Euclidean Absolute are two distinct figures which is a consequence of the Euclidean polarity being singular. In terms of ray and axis coordinates, the induced Euclidean polarity is respectively given by, Fo = 3 (11) YO = (12) and the corresponding polar relations are P = 0 p = =(13) ,. .l P 0 p = YO P = = (14) I3 P P where the 3xl arrays p, p P, P,0 0 are introduced, I OT (15) S= 01 02 03 = 23 P31 12T (15) [T T T6) P = 23 P31 P12 =0 01 02 031 (16) 0 = [0 0 0Q]T. (17) Considering firstly (14), it may be shown that the polar of any line P not on the plane at infinity is in fact a line on the plane at infinity, p = [0 0 0 P23 P31 P12]T Let the line P be the meet of two finite planes, P = JXYI. The polar line p is equivalent to the join of the two poles of X and Y which are points on the plane at infinity, p = X Y and thus represents a line on the plane at infinity. This type of reasoning is not effective for (13). Let line p be the join of two points p = xyl and the polars of x and y are both the plane at infinity. Since the meet of any plane with itself vanishes identically, this does not yield the line given by (13). It is not clear why this synthetic argument fails. However, it should be noted that (13) was derived from (3.1.13) where it was necessary to delete a common factor E before e was set to zero to yield (13). Alternatively, (13) may be derived directly from (14) be multiplying throughout by A, '0 Ap = (A 18) P which yields the desired result P = (19) 0 It also follows from both (13) and (.14) that the polar of any line on the plane at infinity is indeterminate since its coordinates vanish. Thus the Euclidean polar of the Euclidean polar of a line or screw vanishes. This is exactly equivalent to a double application of the Euclidean polar operator w, where w =0, to a rotor or motor (line or screw) and was invented by Clifford [1873] in the development of biquaternions. For elliptic space, Clifford used the polar operator w, where W2=1, which is equivalent to (.3.2.19) and (3.2.20). Clifford did not discuss hyperbolic geometry in 