Bicubic Polar Subdivision
K. Kar6iauskas
University of Vilnius
and
J. Peters
University of Florida
We describe and analyze a subdivision scheme that generalizes bicubic spline subdivision to control
nets with polar structure. Such control nets appear naturally for surfaces with the combinatorial
structure of objects of revolution and at points of high valence when combined with Catmull
Clark subdivision. The resulting surfaces are C2 except at isolated extraordinary points where
the surface is C1 and the curvature is bounded.
Categories and Subject Descriptors: []:
General Terms:
Additional Key Words and Phrases: Subdivision, polar layout, bicubic, CatmullClark, curvature
continuity
1. INTRODUCTION
Polar control nets (Figure 1) capture the combinatorial structure
of objects of revolution and are therefore more natural than the
allquads layout of CatmullClark subdivision [CC78] at points
of high valence (see e.g. Figure 2). We define and analyze a bi
nary subdivision scheme that, just like CatmullClark subdivision,
generalizes the refinement rules of uniform cubic splines but to
polar layout.
Formally, a control net has polar layout [KP06] if it consists of
extraordinary mesh nodes surrounded by triangles, and of quadri
laterals that have nodes of valence four (unless they are on the
global boundary). The extraordinary mesh nodes need only be
separated by one layer of nodes of valence four as shown in Fig
ure 8, left. Applying quadtri subdivision [SL03; PS04; SW05]
to polar layout is not a good alternative, since Loop subdivision
also does not cope well with such input meshes (Figure 2). Polar
subdivision differs structurally from tensored univariate schemes
Sradial circular
A
ci 1
Ci,2
it1
Fig. 1. Polar control net near an extraordinary
point (left) and its refinement (right) under sub
division. The control points cij have subscripts
i indicating (modulo the valence n) the direction
and subscripts j indicating the radial distance to
the extraordinary point cio. Only the radial, not
the circular direction is refined.
with singularities, e.g. [MWW01], in that the number of neighbors of the extraordinary point does not double with
each polar subdivision step but stays fixed. Quadrilaterals in a polar net are not split and the control net refines only
Permission to make digital/hard copy of all or part of this material without fee for personal or classroom use provided that the copies are not made
or distributed for profit or commercial advantage, the ACM copyright/server notice, the title of the publication, and its date appear, and notice is
given that copying is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior
specific permission and/or a fee.
@ 20YY ACM 07300301/20YY/01000001 $5.00
ACM Transactions on Graphucs, Vol V, No N, Month 20YY, Pages 10??
2 K. Kardiauskas and J. Peters
Fig. 2. Wrinkle removal on an Easter Island head (valence 20). (from left to right) CatmullClark subdivision, Loop subdivision (quad facets are
split), control net, colorcoded rings of the polar subdivision surface, polar subdivision surface.
towards the extraordinary point (Figure 1). Therefore, the polar control net does not serve the function of approxi
mating the surface ever more closely by smaller facets, but simply defines it as a Bspline mesh growing towards the
extraordinary point (see, however, Section 4, Control Nets).
Compared to [KMP06], the more localized computation of the present scheme results in a more localized curvature
distribution and higher curvature fluctuations, but has the advantage of simple rules. At the extraordinary point, the
curvature of surface generated by the proposed scheme is only bounded while [KMP06] generates C2 surfaces.
2. POLAR REFINEMENT RULES
We need only explain how to refine the net immediately connected to extraordinary mesh nodes since the remaining
net is a standard bicubic Bspline control net. (For the layer of quadrilaterals adjacent to the triangles, we interpret the
triangles as degenerate quadrilaterals with one edge collapsed.)
Fig. 3. Refinement stencils for binary polar subdivision.
To obtain leading eigenvalues 1, 1, 1, , it suffices to have special rules only at the extraordinary mesh node
and its direct neighbors (Figure 3). The two regular rules are the subdivision rules for univariate uniform cubic splines.
The extraordinary rules have the weights
a 1 := p P 1 := Yk : (C +)2+ (ck3), c cos (2 ) (1)
4' 2' n(B 2 8n n 2n n n
Here, we chose 3 = 1/2 to emphasize convexity at the extraordinary point, since this is likely the dominant scenario
for polar meshes. Section 4, Convexity and Valence discusses the role of 3 in more detail.
A useful property of polar surfaces is that the valence can be changed by interpreting each circular ring of coefficients
as the control polygon of a cubic spline curve. To avoid a special discussion of low valences, we uniformly insert knots
in the circular spline curves when n E {3, 4, 5}, doubling the valence. That is, we can assume n > 6 in the following.
ACM Transactions on Graphics, Vol V No N, Month 20YY
Bicubic Polar Subdivision 3
Fig. 4. (left) Layers 0 through 5 (generated by one subdivision of layers 0 through 3) define (middle) one piecewise bicubic doublering. (right)
Consecutive doublerings join smoothly and, unlike CatmullClark subdivision, without Tcorers.
3. PROPERTIES BY CONSTRUCTION
Let c be the control point of the ith sector and the jth layer as indicated in Figure 1. The central node is considered
split into n copies c', each weighted by 1/n. Then the vector of control points
Cm := (. ., C l, c, Ci, ..) 4n
is refined by a subdivision matrix with blockcirculant structure: cm+1 = Acm,
Ao o A(1 0 00 1 o
A : A ... An, .T2 T1. 4 Ao: A :o 1,...n 1,
A1A .0 o ,Ai o o
AA8n 4 8 0 A11
A1 ... An Ao 0 0 0 0
that can be blockdiagonalized by Discrete Fourier Transform A, := k= .. A .. : exp (2 ), so that
the eigenanalysis is pleasantly simple.
LEMMA 1. For generic input data, the limit surface ofbicubic polar subdivision is C2 except at isolated extraor
dinarypoints where the surface is C1 and the curvature bounded.
PROOF. As illustrated in Figure 4, control point layers 1 through 5 define two rings of bicubic splines (Figure 4
middle). This doublering is C2 since it corresponds to a regular (periodic) tensorproduct spline. As in CatmullClark
subdivision, consecutive doublerings join C2. For n > 5,
if i G{l, n 1}
1 (a1a 00\ 700 000\
ain if {3, n3} andA 1 o A o (2)
,t if i G f3,n \ 3 } (2)
0, if i > 3 and i < n 3.
The eigenvalues ofAo are 1, , 0 and the eigenvalues of Ai for i 1,.. ., n 1, are := , 0,0. In
particular, A1 = 1 and (A)2 A) 2 2 as is required for bounded curvature.
Since the eigenvector of matrix A1 for A1 is (0, 1, 2, 3)t, the subdominant eigenvectors of A are the coordinates
of
v= (...,r ,r r r ,r ,...), rs := k ] i= 1,...,n, k 0,1,2,3.
The control net v defines the characteristic map (Figure 4, middle) [Rei95], whose regularity and injectivity are easily
verified. The eigenvectors corresponding to the eigenvalue 1/4 are from Fourier blocks 0, 2 and n 2 and they are not
generalized eigenvectors. Explicitly, for use in Section 4, the eigenvectors v2k to the eigenvalue of Ak for Fourier
index k e {0, 2} are
1
ACM Transactions on Graphics Vol V No N Month 2YY v (1,4,6, (3)
ACM Transactions on Graphics, Vol V, No N, Month 20YY
4 K. Kardiauskas and J. Peters
Together with the curvature bounded spectrum, this implies curvature boundedness as claimed. [
LEMMA 2. The limit extraordinary point is
1 n 4(1 3)
ijcoo + (1 3l) n cil il : 3
i= 1
PROOF. We choose the representation A e R3"+ 3l"+1 of the subdivision operator where we do not replicate the
central node coo:
Ia a, ... _a,
a, Ao .. A,1 a [ 0 0] 0 n
A := n A i 0: 4 l 0 ) Ai: 0 i=,...,n1.
ac : [1 O, ', 0]t o o oo
a, ... A,1 A 0
We can directly check that the left eigenvector of A with respect to the dominant eigenvalue 1 is
[1/3+3 e,e,a...,e]t, e: a 0, ].
1'3+a n(l, + a)'
The claim follows (see [DKT98], Appendix A) since the entries sum to 1. [
4. DISCUSSION
This section discusses some alternative schemes, the meaning of control polyhedra, integration with CatmullClark
subdivision and adjustment of valence and convexity.
Alternative Schemes.
The bicubic polar subdivision algorithm has spe
cial rules for both the new central node and its di
rect neighbors. Choosing symmetric special rules n
only for the central node does not yield appropriate a a 7 7 7
degrees of freedom for smoothness. Specifically, n Yi
forcing a double subdominant eigenvalue by tun
ing only the rules for the central node, leads to one
single subdominant eigenvector for n > 3; only
for n 3, there exist rules to generate C1 surfaces Fig. 5. Refinement stencils for a ternary polar subdivision (split
with a spectrum suitable for bounded curvature (but ting into three in the radial direction) where/3 : a := 
only a single subsubdominant eigenvector). So, a 4 + 2c )( 2
direct polar analogue of CatmullClark subdivision '" : (5 + 2c^) (1 + K') .
fails and the question arises whether a ternary po
lar subdivision scheme analogous to [Loo02], is advantageous. We derived such a variant for comparison (see
Figure 5). The weights 7k are nonnegative and the scheme satisfies all the constraints on the leading eigenvalues
(1, 3, 3, 9, 9, ) and eigenvectors for curvature boundedness. However, the resulting surfaces did not look better than
those of the proposed binary subdivision.
ACM Transactions on Graphics, Vol V, No N, Month 20YY
Bicubic Polar Subdivision 5
Control Nets.
Subdivision surfaces can either be viewed as refining a control net or as
generating a sequence of surface rings converging to the extraordinary
point [Rei95]. The first serves intuition if the control net outlines the
shape, the second is preferred for exact evaluation, computing and anal
ysis. Both CatmullClark subdivision and polar subdivision admit the
two views but differ in their bias. To see this, define a Tcorner to be the
location where an edge between two distinct polynomial patches meets
the midpoint of an edge of a third. With each refinement, CatmullClark
subdivision generates Tcorers between the patches of adjacent surface
rings (Figure 6, left top). Polar subdivision does not generate Tcorers
since the control net refines only towards the extraordinary point (Figure
1,top right). However, if we want that the control net play the additional
role of a piecewise bilinear faceted approximation converging to the un
derlying surface, we have to split the quadrilaterals into four, generating
Tcorers in the control polyhedron (Figure 6, right bottom). Reflecting
this bias, CatmullClark subdivision is usually illustrated by a sequence
of control nets (Figure 1,left bottom) and polar surfaces by a sequence of
surface rings.
Combining CatmullClark and polar Subdivision.
Figure 12 shows how a design can combine CatmullClark and polar lay
out. By treating the triangular facets as degenerate quadrilaterals, so that
no 3valent vertices are created, we can apply CatmullClark subdivi
sion everywhere in the first step (Figure 7) and thereby separate polar
and CatmullClark extraordinary mesh nodes. Only the polar extraor
dinary mesh node requires its own rule. For a range of 3, the choice
a := 1 yields good shapes, and the extraordinary point does
not change from that computed in Lemma 2. If then a
This first step also doubles the valence n of the polar extraordinary mesh
node. This is no problem and, in fact, can be beneficial as we will see
next.
Convexity and Valence.
Decreasing the parameter 3 in (1) pulls the surface closer to the extraor
dinary mesh node. Recently [GU06] documented how such straightfor
patches
control
facets
CatmullClark polar
Fig. 6. Layout of patches and control poly
hedron for CatmullClark subdivision (left)
and polar subdivision (right). While the T
comers (left top) are intrinsic (the coarser
patch is C" at the Tcomer), the Tcomers
(right bottom) are optional and not part of
the polar control net.
Fig. 7. Separating CatmullClark and polar
extraordinary mesh nodes.
ward manipulation results in a limit surface in the desirable region of a 'shape chart': decreasing 3 emphasizes
convexity. We therefore chose 3 := 1/2 (see Figure 10) over 3 := 5/8 even though the latter yields nonnegative
weights k = n (1 + cn)(1 + 2c )2 > 0. Table I shows the effect of 3 on the subsubdominant eigenvector v2o of
(3), that determines the shape in the convex setting, and its second difference Av20. For3 = 1/2, the sector partition
curves are quadratic and have a more pronounced curvature than for3 = 5/8. We also observed that increasing the
valence by knot insertion improves the curvature distribution for convex neighborhoods (see e.g. Figure 10). This is
ACM Transactions on Graphics, Vol V, No N, Month 20YY
3 3/8 4/8 5/8 6/8
v20 (1, 5, 29, 68)t (1, 2, 11, 26)t (1, 1,5, 12)t (1, 1/2, 2, 5)t
A2v20 (18, 15)t (6,6)t (2,3)t (0,3/2)t
Table I. Coefficients of v20, the eigenvector of the zeroth Fourier mode to the eigenvalue 1/4.
6 K. Kardiauskas and J. Peters
due to the increased symmetry of v20 and the fact that, if a curve is C1 at the central point and opposite curve segments
are mirror images, then the curve is C2.
5. CONCLUSION
The algorithm just defined and analyzed is a polar version of bounded subdivision as pioneered by Sabin [Sab91] and
a polar cousin of CatmullClark subdivision. Its simplicity, and the fact that the output consists of bicubic patches,
recommends it as a useful addition to CatmullClark subdivision, allowing the designer more freedom where high
valence and polar layout are natural.
ACKNOWLEDGMENTS
This work was supported in part by NSF Grants DMI0400214 and CCF0430891. Ashish Myles implemented the
algorithm.
REFERENCES
E. Catmull and J. Clark. Recursively generated Bspline surfaces on arbitrary topological meshes. Computer AldedDesign, 10:350355, 1978.
Tony DeRose, Michael Kass, and Tien Truong. Subdivision surfaces in character animation. In Michael Cohen, editor, 1998, Computer
Graphics Proceedings, pages 8594, 1998.
I. Ginkel and G. Umlauf. Loop subdivision with curvature control. In A. Scheffer and K. Polthier, editors, Proceedings oj ' of Graphics
Processing (SGP), June 2628 2006, Caghan, Italy, pages 163172. ACM Press, 2006.
K. Karciauskas, A. Myles, and J. Peters. A C2 polarjet subdivision. In A. Scheffer and K. Polthier, editors, Proceedings oj ] of Graphics
Processing (SGP), June 2628 2006, Caghan, Italy, pages 173180. ACM Press, 2006.
K. Karciauskas and J. Peters. Surfaces with polar structure. Computing, pages 18, 2006. to appear.
C. Loop. Smooth ternary subdivision of triangle meshes. In Curve and .. Fitting, SaintMalo, volume 10(6), pages 36, 2002.
G. Morin, J.D. Warren, and H. Weimer. A subdivision scheme for surfaces of revolution. Comp Aided Geom Design, 18(5):483502, 2001.
J6rg Peters and LeJeng Shiue. Combining 4 and 3direction subdivision. ACM Trans. Graph, 23(4):9801003, 2004.
U. Reif. A unified approach to subdivision algorithms near extraordinary vertices. Comp Aided Geom Design, 12:153174, 1995.
Malcolm Sabin. Cubic recursive division with bounded curvature. Curves and .. pages 411414, 1991.
Jos Stam and Charles T. Loop. Quad/triangle subdivision. Comput. Graph. Forum, 22(1):7986, 2003.
Scott Schaefer and Joe D. Warren. On C2 triangle/quad subdivision. ACM Trans. Graph, 24(1):2836, 2005.
ACM Transactions on Graphics, Vol V, No N, Month 20YY
Bicubic Polar Subdivision 7
Fig. 8. Mirrored 16sided pyramid. (left) control mesh dij; rendering of the subdivision surface (three light sources) using (middle) CatmullClark
subdivision, (right) polar bicubic subdivision (with Gausscurvature shaded inset).
Fig. 9. Intended ripples: (left) Control net (middle) CatmullClark subdivision; (right) polar subdivision.
control net n = 6 =5 n = 12 = n = 6. = 1 n = 12 =
Fig. 10. Analysis of the effect of changing 3 and n on the (everywhere positive) Gausscurvature of a capped cylinder.
ACM Transactions on Graphics, Vol V, No N, Month 20YY
I ? I
1 2
8 K. Karbiauskas and J. Peters
Fig. 11. (left) Sample meshes; (middle) bicubic surface rings; (right) Polar subdivision surfaces.
Fig. 12. Combining bicubic subdivision with extraordinary rules: CatmullClark rules apply where n 7 4 quadrilaterals meet; polar rules apply
where triangles meet (cf. Figure 7).
ACM Transactions on Graphics, Vol V, No N, Month 20YY
