Bicubic Polar Subdivision
K. Kar
ˇ
ciauskas
Vilnius University
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 in subdivision meshes. The resulting
surfaces are C
2
except at a finite number of isolated points where the surface is C
1
and the
curvature is bounded.
Categories and Subject Descriptors: I.3.5 []: Computational Geometry and Object Modeling; J.6 []: Computer-Aided Engineering
General Terms: Algorithms
Additional Key Words and Phrases: Subdivision, p olar layout, polar net, bicubic, Catmull-Clar k,
curvature continuity
1. INTRODUCTION
circularradial
A
1
2
i−1
i+1
c
i,1
c
i,2
Fig. 1. Polar control net near an extraordinary
point (left) and its refinement (right) under sub-
division. The control points c
ij
have subscripts
i indicating (modulo the valence n) the direction
and subscripts j indicating the radial distance to
the extraordinary point c
i0
. Only the radial, not
the circular direction is refined.
Polar control nets (Figure 1) capture the combinatorial structure
of objects of revolution and are therefore more natural at points
of high valence (see e.g. Figure 2) than the all-quads layout fa-
vored by Catmull-Clark subdivision [Catmull and Clark 1978].
Correspondingly, we define and analyze in the following a binary
subdivision scheme that, just like Catmull-Clark subdivision, gen-
eralizes the refinement rules of uniform cubic splines – but for the
layout of a polar net.
Formally, a control net without boundary is a polar net
[Karˇciauskas and Peters 2007] if it consists of extraordinary mesh
nodes surrounded by triangles, and of quadrilaterals that have
nodes of valence four. The extraordinary mesh nodes need only
be separated by one layer of nodes of valence four as illustrated
in Figure 7, left. Applying quad-tri subdivision [Stam and Loop
2003; Peters and Shiue 2004; Schaefer and Warren 2005] to a
polar net 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 with singularities, e.g. [Morin et al. 2001], in that the number of neighbors of the extraordinary point does
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ACM Transactions on Graphics, Vol. V, No. N, Month 20YY, Pages 1–0??.
2 · K. Kar
ˇ
ciauskas 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.
not double with each polar subdivision step but stays fixed. Quadrilaterals in a polar net are not split and the control
net refines only towards the extraordinary point (Figure 1). Therefore, the polar control net does not, off hand, serve
the function of approximating the surface ever more closely by smaller facets. However, the resulting surface as a
single B-spline mesh growing towards the extraordinary point, i.e. the surface does not have the cascading sequence
of T-corners intrinsic to Catmull-Clark surfaces. Section 4, Control Nets, explains this in detail.
Compared to [Karˇciauskas et al. 2006], the more localized computation of bicubic polar subdivision results in a
more localized curvature distribution. At the extraordinary point, the curvature of surfaces generated by bicubic polar
subdivision is only bounded but need not be continuous while the algorithm in [Karˇciauskas et al. 2006] generates C
2
surfaces. The present scheme has, however, the advantage of simpler rules without visibly sacrificing good shape.
2. POLAR REFINEMENT RULES
Apart from the extraordinary mesh nodes, the polar net defined in the introduction, is a standard bicubic B-spline con-
trol net. For the layer of quadrilaterals adjacent to the triangles, we interpret the triangles as degenerate quadrilaterals
with one edge collapsed. It is easy to check, for example by conversion to B´ezier form, that this interpretation does
not result in singularities in the quadrilateral layer. In order to map a polar net to a refined polar net, we will refine the
bicubic spline net only in the radial direction (cf. Figure 1).
α
n
α
n
α
n
1 − α
1 − β
γ
i+1
γ
i−1
γ
i
1
8
6
8
1
8
1
2
1
2
Fig. 3. Refinement stencils for binary polar subdivision.
As is typical for subdivision algorithms, we need only explain how to refine the polar net immediately connected
to extraordinary mesh nodes. To obtain leading eigenvalues 1,
1
2
,
1
2
,
1
4
,
1
4
,
1
4
, 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
ACM Transactions on Graphics, Vol. V, No. N, Month 20YY.
Bicubic Polar Subdivision · 3
univariate uniform cubic splines. The extraordinary rules have the weights
α := β −
1
4
, β :=
1
2
, γ
k
:=
1
n
β −
1
2
+
5
8
c
k
n
+ (c
k
n
)
2
+
1
2
(c
k
n
)
3
, c
k
n
:= cos
2πk
n
. (1)
Here, we chose β = 1/2 to emphasize convexity at the extraordinary point, since this is likely the dominant scenario
for polar meshes. This choice is also reasonable for saddles (Figure 9). Section 4, Convexity and Valence discusses
the role of β 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 and double the valence when n ∈ {3, 4, 5}. That is, we may assume n ≥ 6 in the
following.
3. PROPERTIES BY CONSTRUCTION
Let c
m
i,j
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
m
i0
, each weighted by 1/n. Then the vector of control points
c
m
:= (. . . , c
m
i0
, c
m
i1
, c
m
i2
, c
m
i3
, . . .) ∈ R
4n×4n
,
is refined by a subdivision matrix with block-circulant structure: c
m+1
= Ac
m
,
A :=
A
0
A
1
... A
n−1
A
n−1
A
0
... A
n−2
.
.
.
.
.
.
.
.
.
A
1
... A
n−1
A
0
∈ R
4n×4n
, A
0
:=
1−α
n
α
n
0 0
1−β
n
γ
0
0 0
1
8n
3
4
1
8
0
0
1
2
1
2
0
, A
i
:=
1−α
n
α
n
0 0
1−β
n
γ
i
0 0
1
8n
0 0 0
0 0 0 0
, i = 1, . . . , n − 1,
that can be block-diagonalized by Discrete Fourier Transform
ˆ
A
i
:=
P
n−1
k=0
ω
ik
n
A
k
, ω
ℓ
n
:= exp
2π ℓ
√
−1
n
, so that
the eigen-analysis is pleasantly simple.
LEMMA 1. For generic input data, the limit surface of bicubic polar subdivision is C
2
except at isolated extraor-
dinary points where the surface is C
1
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 C
2
since it corresponds to a regular (periodic) tensor-product spline. As in Catmull-Clark
subdivision, consecutive double-rings join C
2
. For n > 5,
ˆγ
i
=
1
2
, if i ∈ {1, n − 1}
1
4
, if i ∈ {2, n − 2}
1
16
, if i ∈ {3, n − 3}
0, if i > 3 and i < n − 3.
, and
ˆ
A
0
=
1−α α 0 0
1−β β 0 0
1
8
3
4
1
8
0
0
1
2
1
2
0
!
,
ˆ
A
i
=
0 0 0 0
0 ˆγ
i
0 0
0
3
4
1
8
0
0
1
2
1
2
0
!
. (2)
The eigenvalues of
ˆ
A
0
are 1,
1
4
,
1
8
, 0 and the eigenvalues of
ˆ
A
i
for i = 1 , . . . , n − 1, are ˆγ
i
:=
P
n−1
k=0
ω
ik
n
γ
k
,
1
8
, 0, 0. In
particular, λ
1
= ˆγ
1
and (λ
1
)
2
= λ
2
= ˆγ
2
as is required for bounded curvature.
Since the eigenvector of matrix
ˆ
A
1
for λ
1
=
1
2
is (0, 1, 2, 3)
t
, the subdominant eigenvectors of A are the coordinates
of
v = (. . . , r
i−1
3
, r
i
0
, r
i
1
, r
i
2
, r
i
3
, r
i+1
0
, . . .), r
i
k
:= k
h
cos i
2π
n
sin i
2π
n
i
, i = 1, . . . , n, k = 0, 1, 2, 3. (3)
The control net v defines the characteristic map (Figure 4, middle) [Reif 1995], whose regularity and injectivity
are easily verified [Peters and Reif 1998; Umlauf 1999]. 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
ACM Transactions on Graphics, Vol. V, No. N, Month 20YY.
4 · K. Kar
ˇ
ciauskas and J. Peters
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-corners.
eigenvectors v
2k
to the eigenvalue
1
4
of
ˆ
A
k
for Fourier index k ∈ {0, 2} are
v
20
:= (1 + 3b, 1, 7 + 3b, 16 + 6b )
t
, b :=
1
4β − 1
, v
22
:= (1, 4, 6 , 14)
t
. (4)
Together with the curvature bounded spectrum, this implies curvature boundedness as claimed.
LEMMA 2. The limit extraordinary point is
ηc
00
+ (1 − η)
1
n
n
X
i=1
c
i1
η :=
4(1 − β)
3
.
PROOF. We choose the representation
¯
A ∈ R
3n+1×3n+1
of the subdivision operator where we do not replicate the
central node c
00
:
¯
A :=
1−α a
r
... a
r
a
c
¯
A
0
...
¯
A
n−1
.
.
.
.
.
.
.
.
.
a
c
...
¯
A
n−1
¯
A
0
a
r
:= [
α
n
, 0, 0]
a
c
:= [1 − β,
1
8
, 0]
t
¯
A
0
:=
γ
0
0 0
3
4
1
8
0
1
2
1
2
0
¯
A
i
:=
γ
i
0 0
0 0 0
0 0 0
, i = 1 , . . . , n − 1 .
We can directly check that the left eigenvector of
¯
A with respect to the dominant eigenvalue 1 is
[
1 − β
1 − β + α
,ℓ
ℓ
ℓ,ℓ
ℓ
ℓ, . . . , ℓ
ℓ
ℓ]
t
, ℓ
ℓ
ℓ := [
α
n(1 − β + α)
, 0, 0].
The claim follows (see [DeRose et al. 1998], Appendix A) since the entries sum to 1.
Since [0, 1, 0, 0] is a left eigenvector to
ˆ
A
1
, the normal direction at the extraordinary point is simply
(
P
n
i=1
cos i
2π
n
c
i0
) × (
P
n
i=1
sin i
2π
n
c
i0
).
4. DISCUSSION
This section discusses some alternative schemes, the meaning of control polyhedra and adjustment of valence and
convexity.
Alternative Schemes.
The bicubic polar subdivision algorithm has special rules for both the new central node and its direct neighbors.
Choosing symmetric special rules only for the central node does not yield appropriate degrees of freedom for smooth-
ness. Specifically, forcing a double subdominant eigenvalue by tuning only the rules for the central node, leads
to one single subdominant eigenvector for n > 3; only for n = 3, do there exist rules to generate C
1
surfaces
with a spectrum suitable for bounded curvature. So, a direct polar analogue of Catmull-Clark subdivision fails
and the question arises whether a ternary polar subdivision scheme, analogous to [Loop 2002], is advantageous.
ACM Transactions on Graphics, Vol. V, No. N, Month 20YY.
Bicubic Polar Subdivision · 5
α
n
α
n
α
n
1 − α
1 − β
γ
i+1
γ
i−1
γ
i
10
27
16
27
1
27
4
27
19
27
4
27
Fig. 5. Refinement stencils for a ternary polar subdivision (splitting into three in the radial direction) where β :=
38
81
, α := β −
1
9
, γ
k
:=
4
81n
`
5 + 2c
k
n
´`
1 + c
k
n
´
2
.
We derived such a variant for comparison (see Figure 5). The weights γ
k
are non-negative and the scheme sat-
isfies all the constraints on the leading eigenvalues (1,
1
3
,
1
3
,
1
9
,
1
9
,
1
9
) and eigenvectors for curvature boundedness.
patches
control
facets
polar
Catmull-Clark
Fig. 6. Layout of patches and control
polyhedron for Catmull-Clark subdivision
(left) and polar subdivision (right). The T-
corners in Catmull-Clark (left top) are in-
trinsic (the coarser patch is C
∞
at the T-
corner). The T-corners in the refined poly-
hderon (right bottom) are optional and not
part of the polar net.
However, the resulting surfaces did not look better than those of the pro-
posed binary subdivision.
Control Nets and Surface Rings.
Subdivision surfaces can either be viewed as refining a control net or as
generating a sequence of surface rings converging to the extraordinary
point [Reif 1995]. The first serves intuition if the control net outlines the
shape, the second is preferred for exact evaluation, computing and analy-
sis. 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-corners between the patches of adjacent surface
rings (Figure 6, left top). Polar subdivision does not generate T-corners
(Figure 6, right top) since the control net refines only towards the extraor-
dinary point (Figure 1). One approach for generating a faceted approxi-
mation converging to the underlying surface is to split the quadrilaterals
of the polar net at each refinement into four and leave the triangles un-
touched. This yields T-corners in the faceted approximation (Figure 6,
right bottom). Reflecting the bias towards presenting a mesh without T-
corners versus obtaining a surface without T-corners, Catmull-Clark sub-
division is usually illustrated by a sequence of control nets (Figure 1,left
bottom), hiding the surface T-corners, while polar surfaces are preferably
introduced as a sequence of surface rings.
Convexity and Valence.
Decreasing the parameter β in (1) pulls the surface closer to the extraor-
dinary mesh node. Recently [Ginkel and Umlauf 2006] documented how
such straightforward manipulation results in a limit surface in the desir-
able region of a ‘shape chart’ [Karciauskas et al. 2004]: decreasing β emphasizes convexity. We therefore chose β :=
1/2 (see Figure 10) over β := 5/8 even though the latter yields non-negativeweights γ
k
=
1
8n
(1+c
k
n
)(1+2c
k
n
)
2
≥ 0.
Table I shows the effect of β on the subsubdominant eigenvector v
20
of (4), that determines the shape in the convex
setting, and its second difference ∆v
20
. For β = 1/2, the sector partition curves are quadratic and have a more pro-
nounced curvature than for β = 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 due to the increased symmetry of v
20
and
the fact that, if a curve is C
1
at the central point and opposite curve segments are mirror images, then the curve is C
2
.
ACM Transactions on Graphics, Vol. V, No. N, Month 20YY.
