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Title: Bicubic polar subdivision
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Title: Bicubic polar subdivision
Alternate Title: Department of Computer and Information Science and Engineering Technical Report
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Language: English
Creator: Peters, J.
Karciauskas, K.
Publisher: Department of Computer and Information Science and Engineering, University of Florida
Place of Publication: Gainesville, Fla.
Copyright Date: 2007
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Bicubic Polar Subdivision

K. Kar6iauskas
University of Vilnius
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, Catmull-Clark, curvature


Polar control nets (Figure 1) capture the combinatorial structure
of objects of revolution and are therefore more natural than the
all-quads layout of Catmull-Clark subdivision [CC78] at points
of high valence (see e.g. Figure 2). We define and analyze a bi-
nary subdivision scheme that, just like Catmull-Clark 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 quad-tri 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


ci 1
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

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2 K. Kardiauskas and J. Peters

Fig. 2. Wrinkle removal on an Easter Island head (valence 20). (from left to right) Catmull-Clark subdivision, Loop subdivision (quad facets are
split), control net, color-coded 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 B-spline 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.

We need only explain how to refine the net immediately connected to extraordinary mesh nodes since the remaining
net is a standard bicubic B-spline 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.
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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 double-ring. (right)
Consecutive double-rings join smoothly and, unlike Catmull-Clark subdivision, without T-corers.

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 block-circulant structure: cm+1 = Acm,
Ao o A(1 0 00 1 o
A : A ... An-, -.T-2 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 block-diagonalized by Discrete Fourier Transform A, := k= .. A .. : exp (2 ), so that
the eigen-analysis 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 double-ring is C2 since it corresponds to a regular (periodic) tensor-product spline. As in Catmull-Clark
subdivision, consecutive double-rings join C2. For n > 5,

if i G{l, n 1}
1 (a1a 00\ 700 000\
ain if {3, n-3} 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
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
ACM Transactions on Graphics Vol V No N Month 2YY v (1,4,6, (3)
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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:
I-a a, ... _a,
a, Ao .. A,1 a [ 0 0] 0 n
A := n A i 0: 4 l 0 ) Ai: 0 i=,...,n-1.
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. [

This section discusses some alternative schemes, the meaning of control polyhedra, integration with Catmull-Clark
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 Catmull-Clark 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 non-negative 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.

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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 Catmull-Clark subdivision and polar subdivision admit the
two views but differ in their bias. To see this, define a T-corner to be the
location where an edge between two distinct polynomial patches meets
the midpoint of an edge of a third. With each refinement, Catmull-Clark
subdivision generates T-corers between the patches of adjacent surface
rings (Figure 6, left top). Polar subdivision does not generate T-corers
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
T-corers in the control polyhedron (Figure 6, right bottom). Reflecting
this bias, Catmull-Clark subdivision is usually illustrated by a sequence
of control nets (Figure 1,left bottom) and polar surfaces by a sequence of
surface rings.
Combining Catmull-Clark and polar Subdivision.
Figure 12 shows how a design can combine Catmull-Clark and polar lay-
out. By treating the triangular facets as degenerate quadrilaterals, so that
no 3-valent vertices are created, we can apply Catmull-Clark subdivi-
sion everywhere in the first step (Figure 7) and thereby separate polar
and Catmull-Clark 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

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-



Catmull-Clark polar

Fig. 6. Layout of patches and control poly-
hedron for Catmull-Clark subdivision (left)
and polar subdivision (right). While the T-
comers (left top) are intrinsic (the coarser
patch is C" at the T-comer), the T-comers
(right bottom) are optional and not part of
the polar control net.

Fig. 7. Separating Catmull-Clark 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 non-negative
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

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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.


The algorithm just defined and analyzed is a polar version of bounded subdivision as pioneered by Sabin [Sab91] and
a polar cousin of Catmull-Clark subdivision. Its simplicity, and the fact that the output consists of bicubic patches,
recommends it as a useful addition to Catmull-Clark subdivision, allowing the designer more freedom where high
valence and polar layout are natural.

This work was supported in part by NSF Grants DMI-0400214 and CCF-0430891. Ashish Myles implemented the

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ACM Transactions on Graphics, Vol V, No N, Month 20YY

Bicubic Polar Subdivision 7

Fig. 8. Mirrored 16-sided pyramid. (left) control mesh dij; rendering of the subdivision surface (three light sources) using (middle) Catmull-Clark
subdivision, (right) polar bicubic subdivision (with Gauss-curvature shaded inset).

Fig. 9. Intended ripples: (left) Control net (middle) Catmull-Clark 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) Gauss-curvature 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: Catmull-Clark 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

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