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Title: Gaussian and mean curvature of subdivision surfaces
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Umlauf, Georg
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Publication Date: March 10, 2000
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Gaussian and Mean curvature of subdivision surfaces*


UFL, CISE TR 2000-01


Jorg Peters**, Georg Umlauf*

March 10, 2000


Abstract

By explicitly deriving the curvature of subdivision surfaces in the extraordinary points, we give an
alternative, more direct account of the criteria necessary and sufficient for achieving curvature continuity
than earlier approaches that locally parametrize the surface by eigenfunctions.
The approach allows us to rederive and thus survey the important lower bound results on piecewise
polynomial subdivision surfaces by Prautzsch, Reif, Sabin and Zorin, as well as explain the beauty of
curvature continuous constructions like Prautzsch's. The parametrization neutral perspective gives also
additional insights into the inherent constraints and stiffness of subdivision surfaces.


1 Introduction

Almost all subdivision algorithms in the current literature achieve tangent continuity but not curvature conti-
nuity. We give a simple characterization of the causes underlying this phenomenon by explicitly expressing
Gaussian and mean curvature in the minimally smooth extraordinary points. This allows us to rederive and
thereby survey the important lower bound results of [Sabin '91, Reif '96, Zorin '98, Prautzsch & Reif '99]
and constructions for curvature continuous piecewise polynomial subdivision algorithms by [Prautzsch '97,
Prautzsch & Umlauf '98b, Reif '98b]. Beyond this we get additional insights into the inherent constraints
and stiffness of such subdivision algorithms. Since a subdivision surface consists of an infinite collection
of polynomial pieces around every extraordinary point one might expect such surfaces to be more flexible
than spline surfaces. However, we will see that the infinite application of the same subdivision rule enforces
strict rules on the piecewise polynomial rings converging towards extraordinary points. For example, the
Jacobian of the subdominant eigenfunctions of a curvature continuous subdivision algorithm must have
lower degree than the Jacobian of the subdivision surface itself.
The paper is organized as follows. With the notation of Section 2, we express in each subdivision step m
the curvatures of the innermost spline ring around a given extraordinary point as Km = (/A2)" 2mf (u, v),
respectively, HH = (p/A2)m f7 (u, v) for scalar constants p < A and rational functions f. The factor p/A2
immediately implies necessary constraints on curvature continuous subdivision surfaces and the weakest
form of curvature smoothness: the principal curvatures of piecewise polynomial C1 subdivision algorithms
are square integrable. Section 4 derives and reviews necessary constraints on subdivision surfaces to be cur-
vature continuous by observing that f should be constant in the limit and equating the degree of numerator
and denominator of f at the extraordinary point. Section 5 reviews Prautzsch's sufficient condition and his
unique construction of a curvature continuous, linear, stationary subdivision algorithm by projection.

Supported by NSF grant CCR-9901894.
CISE, University of Florida, P.O. Box 116120, Gainesville, FL 32611-6120, USA,
e-m ail: [1..., , i , iii, -*I











Jorg Peters and Georg Umlauf


2 Notation and basic facts

In this section we define just the basic notation and facts needed for our analysis; for a formal, more general
and abstract setting, the reader is referred to [Reif '98a] and [Zorin '98].
While our analysis applies to a larger class of subdivision algorithms, we focus in the following on
generalized box-spline subdivision i.. I ii, ii,, that is on affine invariant, symmetric, linear, local, stationary
algorithms that generalize box-spline subdivision and generate (regular) C1 surfaces. In particular, the limit
surface has a piecewise polynomial parametrization and the parametric smoothness between the pieces is
well-known except at a finite number of extraordinary points. An extraordinary point is the limit point of
a minimal subnet of the initial control net under repeated application of the subdivision algorithm. Such
a subnet consists of an n-valent vertex (for a primal subdivision algorithm) or an n-sided facet (for a dual
subdivision algorithm) and just those neighboring control points that determine a surface ring xo around an
n-sided hole by the regular box-spline subdivision rules (Figure 1 left).














Figure 1: Inserting a surface ring x. (light grey) into the hole left by x,-i (dark grey). x. consists of
5 images of 20 \ E. The vector of control points Cm of x. (circles) is obtained by one subdivision step
applied to C m- (dots).

If we arrange the points of the subnet into the column vector Co then each subdivision step transforms
the subnet by applying the same square, stochastic subdivision matrix A. After m applications the result is
the subnet
Cm = AmCo.
The mth subdivision step adds the surface ring x..(u, v) inside the hole left by the (m 1)st subdivision
step (Figure 1 right). The surface rings are box-splines and can therefore be represented as

Xm : {0,..., n 1} x n -4 WI3, X.(u, iv) = B(u, v)Cm,

where B(u, v) is the row vector of the box-spline basis functions and the domain S is either 20 \ O or
2A \ A, with O the unit square and A the unit triangle.
For simplicity we assume that A is diagonalizable with eigenvalues

1 = Ao > A, = A2 > A3 = A4 = A5 >...> 0,
=:A

such that A1 = A2 correspond to the 1st and (n 1)st block, As = A4 (for n > 3) to the 2nd and
(n 2)nd block and As to the Oth block of the Fourier decomposition of A (c.f [Peters & Reif '98]). The
corresponding eigenvectors are denoted by vi, i.e. Av, = Aivi for all i. For subdivision algorithms with
more general subdivision matrices A the reader is referred to [Reif '98a, Zorin '98].










Gaussian and Mean curvature of subdivision surfaces


In terms of multiples pi of the eigenvectors the subnet can be expressed as

Cm = Amvipi, pi E .

Expanded in the i. ,.i. l,,. r,. e* : {0,..., n 1} x 0 -+ R, (u, v) - B(u, v)vi associated with vi the
surface ring x, is of the form

Xm(u,v) = A7 B(u,v)vipi = Aei(u, i'p
i i

A well-known fact of differential geometry (see e.g. [Carmo '76]) is that for any regular surface para-
metrization x the Gauss curvature K and the mean curvature H are
e(u, ) .,(., v) f(u, v)2
v) E(u, v)G(u,v) F(u, v)2
Se(u, v)G(u, v) 2f(u, v)F(u, v) + g(u, v)E(u, v)
2(E(u, v)G(u, v) F(u, v)2)
where x, is the partial derivative of x(u, v) with respect to u
E=xuxu, F=Xux,, G=xx
e =nxL, f = nxU g = nxV,
and n = (x, x x,)/ \, x x, | is the normal. Since x is assumed to be regular, the denominators of (1),
EG F2 = I\, x. are nonzero and we have

S det(x, x,, xuu) det(xu, x,, xv) det(xu,x,, x,,)2
K =
Ix, X x. I'
det (xu, xv xv )(xU x ) 2 det(x, xv,, x )(xX,) + det(xex,, x,,)(xXu )
H =
2Ix, x x I

3 Gauss curvature and mean curvature

In this section, we derive the Gauss curvature and the mean curvature of the limit surfaces of generalized
box-spline subdivision algorithms at extraordinary points. We expand each of the surface rings x, in
terms of the eigenfunctions e*. This approach goes back at least to [Reif '93]. However, in contrast to
[Reif '93] we do not analyze the curvature by parametrizing the limit surface locally as a function over the
subdominant eigenfunctions el and e2 but rather compute the curvature expansion explicitly. That is, we
determine the curvatures K, (u, v) and H, (u, v) of xm (u, v) and then take the limit as m -+ oo.
For simplicity we write x instead of xm in the following. Since the basis functions form a partition of
unity, eo 1 and
Xu = A'i ( 2 P, 4 e5
x, =A (eUpi + eup2) + m ( p + eup4 + euP5) + O(lm),
xI 2 -m (1 2 m + 4 e5 P O t
x, = Am (eu pi + euvp2) + (evP3 + evP4 + v5) o(pm)

Symmetry in u and v yields the analogous terms for x,, xu and x,,. With the abbreviations
ij :. eAe -- e

DPt := det(pi, p- A,2 A .pt, p 8, {v
Pij := det(pi, p2, pi) det(pi, p2, pj),










Jorg Peters and Georg Umlauf


it is now easy to see that

x, x x, = A A212(pi x p2) + o(A2m),
det(x,,x,, x) = A2"m1m det(pi,p2,pi)D. + o(A2m pm").
i=3,4,5
Symmetry yields the analogous terms for det(xU, x,, xi,) and det(xu, x,, x,,) and

SPj (DD, -D DDD) + o(l)
= m i,j=3,4,5A. pI x p +() 2)
T2 I I)L X 1) + 0(1)
All dependencies on m in the equality are either explicit or hidden in the o(1) terms. We note that A12 is
the Jacobi determinant of the characteristic map [Reif '93, Reif '95], which is non-zero if the characteristic
map is regular, and that I [ x p is positive for almost all initial control nets Co. The leading factor of the
expression (2) for the Gauss curvature readily yields the following basic characterization of the curvature
at extraordinary points (c.f. [Reif '93, page 25])
Observation 1 Let A be the subdivision matrix of a regular C1 generalized box-spline subdivision algo-
rithm as defined in Section 2 with eigenvalues 1 > A > pt > ... > 0.
(a) Ifp > A2 then the Gauss curvature at the extraordinary point is infinite.
(b) If p < A2 then the Gauss curvature at the extraordinary point is zero.
(c) If t = A2 then the Gauss curvature at the extraordinary point is bounded by the secondfactor of(2),
but is possibly non-unique.

Examples for (a) are [Catmull & Clark '78, Loop '87, Qu '90], for (b) are [Prautzsch & Umlauf '98a,
Prautzsch & Umlauf '98b] and for (c) are [Sabin '91, Holt '96]. Note the curious combination of tangent
continuity and infinite curvature for the standard algorithms in (a). In case (c), the limit for m -4 oo yields
at the extraordinary point

K= P'i Di Dv -, D D (3)
K AIm4xp.' (3)
i,j=3,4,5
Recall that the factor (DJD,, D ,DV)/A12 is a rational function in u and v. In order for the Gauss
curvature to be well-defined at the extraordinary point rather than multi-valued or divergent, K must be
constant. Since the Pij depend on the initial net, they can be arbitrary except for Pij = Pji and each
summand has to be constant. We conclude that the eigenfunctions el,..., e5 must satisfy the six partial
differential equations:

DDD, 2DJD~ + DD~ A42 constij, fori, j E {3,4,5}, j > i,
D D (D )2 = 2 .constii, for i= 3,4,5.
Lemma 2 The limit surface of a generalized box-spline subdivision ui5,. .a ih,, with A12 / 0 has continu-
ous, for almost all initial nets non-zero, Gauss curvature at the extraordinary point if and only if p = A2
and the dirltrential equations (4) hold.
Similarly, with Pkl := det(pi,p2,pi)( p, p') and Sk = 1/2 for k = I and 1 otherwise, the mean
curvature is
S Pikl (klD,(ee) D,(eel, + + ee) + EklDU(eke,)) + o(l)
H A^ m lp xk, 1=,2,p k+
H_ ,(5)
T2 1) L X 1) + 0(1)










Gaussian and Mean curvature of subdivision surfaces


The expression for Hm yields Observation 1, with Gauss curvature replaced by mean curvature. For p = A2
we get bounded, not forcibly zero, but possibly non-unique mean curvature

P ikl kD (ee) D, (e e, + e k) EkD (ek e)
H ) + (6)
gP= X- l--l' X I1 A12
k,1=1,2, E>1
In analogy to Lemma 2 the necessary and sufficient conditions for a mean-curvature continuous limit surface
require that nine partial differential equations hold:

kl D,(ee) Dk(e e, + e e,) + klDvA(ee) = Aa2 constik, (7)

fori = 3,4,5, k, = 1,2, k > I.
Since the principal curvatures at a point on the mth surface ring are

1,2 = :Hm ,+ VH Km,

(2) and (5) imply that r,' and rm converge like O(pm/A2m,) for m o0.

Observation 3 The limit surface ofa generalized box-spline subdivision l ..i. ah,,i with A12 / 0 is curva-
ture continuous at the extraordinary point if p = A2 and the ditti'rntial equations (4) and (7) hold.

Since f dxm = O(A22m) and p < A

12 m2 2dxm= 0( O2m"/A2" ) <= oc.

This immediately implies an interesting fact derived for general LP spaces in [Reif & Schroder '00].

Observation 4 The principal curvatures of the limit surface of a generalized box-spline subdivision algo-
rithm are square integrable.


4 Lower bounds on the degree

We now take a look at the important lowerbound results of [Sabin '91, Reif '96, Zorin '98, Prautzsch & Reif '99].
For this it is crucial to distinguish between the apparent or formal degree of box-spline eigenfunctions, pos-
sibly the result of degree-raising, and the true degree denoted by "deg". The true degree is defined to be
the minimal number of non-vanishing derivatives. We focus on the differential equations resulting from the
Gaussian curvature the analysis of the mean curvature yields the same results.
We recall that the left hand side of the differential equations (4) are for i, j = 3,4, 5,

Gij :=DD D 2DD + DDu, j > i,
Gii :=DuDvD (D',)2

Let d = I1 -_ (x,,) denote the total degree, respectively, the bi-degree of a regular box-spline parametrization.
A straightforward count yields for all i, j = 3, 4, 5, j > i, that

for the total degree ,l< (Gij) < 2(2(d 1) + d 2) = 6d 8 and

for the bi-degree ,l .- (Gij) < 2(2d 1 + d 1) = 6d 4,


whereas for the right hand side of (4)










Jorg Peters and Georg Umlauf


the formal total degree of Af2 is 4(2d 2) and

the formal bi-degree of Af2 is 4(2d 1).

This degree mismatch implies the following observation.

Observation 5 The limit surface ofa generalized box-spline subdivision ilg... i dim with total degree

S-_ (x,,) = d and deg(Al2) = 2(d- 1)

is curvature continuous at an extraordinary point P if and only if P is a flat point, i.e. p < A2. The limit
surface of a generalized tensor-product subdivision ,ihg.. mil with bi-degree

I.- x,,) = d and ._(AL2) = 2d- 1

is curvature continuous at an extraordinary point P if and only ifP is aflat point.

In other words, a generalized box-spline subdivision algorithm can only have a curvature continuous limit
surface for p = A2, if the true degree of the Jacobian A12 is less than its formal degree.This is the case, if
either one or both of the following conditions hold:

(i) The true degree of e1 or e2 is less than d.

(ii) The leading terms in the Jacobian A12 cancel.

Since we assume symmetric masks,

1 -i_ (el) = I tl.i (e2) =: d'.

If the subdivision surface is curvature continuous and not flat in the extraordinary point and if condition (ii)
does not apply then d' < d must hold by condition (i). In fact, we compute

for the total degree ,l (Gij) = 2(2d' + d 4) and ,1 -_ (4A2) = 4(2d' 2) and

for the bi-degree ,li (Gij) = 2(2d' + d 2) and ,1 (A42) = 4(2d' 1).

Comparing degrees we find in either case that 2d' = d and arrive at the following observation:

Observation 6 If the leading terms in the Jacobian A12 do not cancel then the limit surface ofa generalized
box-spline subdivision ui ... i di L is curvature continuous and notflat in an extraordinary point only if the
true (I,- .i,..,.,.' of the surface is at least twice the true (I,- .i,..,.,.' of the subdominant ,. ,..." ,, ii,.' el
and e2.

This is consistent with the degree estimate of [Reif '96, Zorin '98]. The central idea of these proofs appears
already as a parting sentence in [Sabin '91]: the surface is viewed as a function over the tangent plane
parametrized by e1 and e2. To have non-zero curvature, it is necessary that the non-tangential component
of the surface is at least quadratic in e1 and e2, i.e. d > 2d'. More generally, if the non-tangential component
of the surface is at least of degree r in e1 and e2 then the surface representation has to be at least of degree
rd'. Since e1 and e2 have to have a minimal degree to form Ck rings, e.g. d' > k + 1 in the tensor-product
case, a lower bound, say r(k + 1), is deduced [Prautzsch & Reif '99].
If, on the other hand, the subdivision surface is curvature continuous and not flat in the extraordinary
point and if condition (i) does not apply then the leading terms of A12 must cancel by condition (ii). Now
we compute

for the total degree .l (Gij) = 2 max{.h .- (A 12) + d 2, 2(d 1) + d 2} = 6d 8 and










Gaussian and Mean curvature of subdivision surfaces


for the bi-degree il .- (Gij) = 2 max{tl t_ (A12) + d 1, 2d 1 + d 1} = 6d 4.

Comparing against t .- (.A 2) = 1 di (A [2) we obtain a counterpart to Observation 6.

Observation 7 If the true degree ofe1 and e2 is not less than d then the limit surface of a generalized
box-spline subdivision i.. 'idi dii is curvature continuous and not flat in an extraordinary point only if the
total degree lt -_ (A12) < 3d/2 2, respectively, the bi-degree il .- (A12) < 3d/2 1.

From these observations, it is evident that the key to curvature continuous subdivision surfaces is the
answer to the following question.

Central Question For what choices of ,.,.,,i,,. i,. ,,' el and e2 is h :_ (A 12) less than 2> 1- I_ (,,) 2for
total degree, respectively, 2 l ._ ( i,,) 1 for bi-degree generalized box-spline subdivision l...i. ri,,,i


5 Curvature continuous subdivision constructions

If we interpret curvature smoothness in the weak sense of L2 integrability, then Observation 4 guarantees
that almost all C1 subdivision algorithms qualify as curvature smooth. If we allow flat spots, then [Prautzsch
& Umlauf '98a, Prautzsch & Umlauf '98b] yield low degree, small mask, curvature continuous subdivision
algorithms. If we want non-zero bounded curvature, we can adapt the leading eigenvalues as in [Sabin '91,
Holt '96]. However, if we want curvature continuity without flat spots the stringent constraints of Lemma
2 apply and a degree-reduced Jacobian in the sense of Observations 6 or 7 is necessary. A trivial example
that satisfies these constraints is the regular case of any C2 box-spline: here e1 and e2 are linear. The
only non-trivial constructions reported to date subdivide a polynomial filling of an n-sided hole obtained
by projection [Prautzsch '97, Reif '98b].
To see in our newly acquired framework why these constructions yield curvature continuous subdivision
surfaces without forced flat points we restate the sufficient conditions derived by Prautzsch [Prautzsch '98].
(Parametrizing the limit surface over the characteristic map, [Reif '98a] concludes also the necessity of
these conditions.)
The .'ffic ient conditions of [Prautzsch '98] state, that in order to be able to solve the differential equa-
tions (4) it suffices that the eigenfunctions e3, e4 and e5 be quadratic polynomials in el and e2:

ei = a(el)2 + biele2 c(e2)2, ai, b, c E R, for i = 3,4,5. (8)

Indeed the derivatives of e', i = 3, 4, 5, are then of the form

e = 2aiele + bi(ee2 ele2) + 2c e2 e,
et = 2ai (e)2 + ele) + b (e e2 + 21..q + e1 e,) + 2ci ((e2)2 + e2e,)

and analogously for ei, e~ and e This yields

Ali = A12(be + 2cie2),
A2i = -A12(2aie + bie2) and
D, = 2l2(ai(e 2 + bie e + c1(e )2).

Substituting Di,, D~ and D, in (3) and (6), the limit terms for the Gauss and the mean curvatures
simplify to

K = p -. fi and H= P p. i
i,j=3,45 L p j i=3,4, 5. X
k.1=1.2.. k>l.










Jorg Peters and Georg Umlauf


where

1\r i f c for k = I = 1
( 4(aicj + ajc) 2bibj fori j f C fork =I
j 4aic (b )2 foori = = jfork
-bi/2 for k :/ I
Since these last expressions for the Gauss and mean curvature are constant, the limit surface is curvature
continuous at the extraordinary point.
Subdivision algorithms that satisfy the sufficient conditions (8) were derived by [Prautzsch '97, Reif
'98b]. We analyze Prautzsch's approach in detail in terms of our necessary and sufficient conditions of
Lemma 2. Here vi and v2 are the eigenvectors to the subdominant eigenvalue A of the Catmull-Clark
algorithm. Then e1 and e2 have true bi-degree 3. Now set e3 = (e)2, e4 = ele2 and e5 = (e2)2 with
control nets wi, i = 3, 4, 5. Furthermore, wi and w2 are the control nets of e1 and e2, respectively, in a
degree-doubled representation. Then the subdivision matrix of Prautzsch's construction for is given by

A = MDM+

where

M := [1,wi,w2,w3,w4, \ D:= diag(1, A A2),, A2 := (Mltm)-1 m.

In this construction the only non-zero eigenvalues of A are 1, A(2-fold), A2(3-fold) corresponding to the
eigenvectors 1, wi,..., w5.


6 Conclusion

We surveyed and restated a number of important recent results concerning the curvature continuity of the
limit surfaces of generalized box-spline subdivision algorithms. The direct computation of K and H sim-
plifies the matter and yields new insights into why the limit surfaces of most subdivision algorithms are not
curvature continuous and what criteria need to be enforced by new constructions.


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