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On the Complexity of Smooth Spline Surfaces

from Quad Meshes

Jorg Peters and Jianhua Fan

March 9, 2009


Abstract
A standard task of CAGD is to determine G1 -connected tensor-product
B-spline patches approximating a quadrilateral mesh whose vertices can
have any fixed valence. This paper derives fundamental relations that suf-
ficiently smooth regular patches have to obey to be part of a C1 complex
and that restrict the choice of G -reparameterizations when patches are
computed by local averaging.
In particular, when one patch is associated with each quadrilateral
facet, and the patch is a polynomial spline of degree bi-3, then the relations
dictate the minimal number and multiplicity of knots: for general data
and geometric fl. .I.lIi-., there must be at least two internal knots per
edge and at least one must be a double knot.


1 Introduction

Quad(rilateral) meshes are used in geometric design and computer graphics,
because they capture symmetries of natural and man-made objects. Smooth
surfaces of degree bi-3 can be generated by applying subdivision to the quad
mesh [C'C -] or, alternatively, by joining a finite number of polynomial pieces
(see e.g. [Pet00]). When quads form a checkerboard arrangement, we can inter-
pret 4 x 4 grids of vertices as control points of bi-3 B-spline surface. Then we
call the central quad ordinary and are guaranteed that adjacent ordinary quad
patches join C2.
The essential challenge comes from covering extraordinary quads, i.e. quads
that have one or more vertices of valence n / 4. While this can be addressed
by recursive subdivision, representing the surface with a finite small number of
patches is often preferable. In particular, to take advantage of parallelism in
the construction, such as GPU acceleration (see e.g. [LS08, NY- ,I i-]), finite,
localized, parallelizable construction steps are needed. That is, the construc-
tion near vertices must dependent only on a fixed, small neighborhood of the
vertex and the remaining construction for each quad depends only on the newly
computed vertex data in a small neighborhood of the quad.















This raises the fundamental question, Q, answered in this paper: what is
the simplest structure (least number of knots) of degree bi-3 spline patches that
allow a general quad mesh, including extraordinary quads, to be converted by
localized operations into a smooth surface with one spline patch per quad?
Section 2 takes a more general view. Here we do not constrain the domain
to be a collection of quadrilaterals or the functions to be polynomial splines.
The relations of Lemma 1, 2 and 3 in main part also do not depend on locality
of the construction but apply to any collection of sufficiently smooth patches
coming together with an unbiased, logically symmetric G1 join (Definition 1,
3). Adding the locality requirement in Section 2.1 allows us to rule out certain
G1 reparameterizations.
In Section 3, we specialize the setting to polynomial tensor-product splines of
degree bi-3. For these, we deduce a lower bound on the number and multiplicity
of knots. We prove that at least two internal knots are required per edge and
at least one must be a double knot to admit a local construction. These lower
bounds are tight since constructions for smooth surfaces (without the shape
defects characterized in Lemma 5) exist for the minimal cases of exactly two
internal knots [FP08] (and for one internal double knot and several single knots).
Together, these lower and upper bounds (algorithms) conclusively settle the
question Q.

1.1 Bi-3 constructions in the literature
Creating C1 surfaces of degree bi-3, i.e. generalizing standard tensor-product B-
spline layout to arbitrary manifold quad meshes with first-order differentiability,
is a classic challenge of CAGD (see e.g. [Bez77, vW86, Pet91]). More recently,
a number of papers appeared that are predicated on the conclusion that simple
constructions are not possible. PC('C. I [PetOO] generates smooth bi-3 surfaces
but requires up to two steps of Catmull-Clark subdivision to separate non-4-
valency vertices. This proves that a 4 x 4 arrangement of polynomial patches per
quad, corresponding to two internal double knots and one single knot, suffices
in principle. However, [PetOla] pointed out that PCC('. I can foce poor shape for
certain higher-order saddles (see also Lemma 5). Similarly, [SWWL04] justifies
a subdivision-like refinement approach with C2 bi-3 tensor-product patches to
obtain approximately smooth surfaces by, correctly as we will see, stipulating that
no finite exact construction is possible. Loop and ". I, i. i. i [I.s "] propose a bi-3
C surface construction with separate tangent patches to convey the impression
of smoothness when applying lighting, a technique cwproposed for computer
graphics in [VPBM01]. Again, the results in this paper justifies the approach
if one restricts each patch to be a cubic polynomial. For the same reason,
Myles et al. [MYP08] perturb a bi-3 base patch near non-4-valency vertices
to obtain a smooth surface suitable for CAD applications with one bi-5 patch
per quad. As mentioned, [FP08] gives an algorithm that constructs smoothly
connected Bezier patches of degree bi-3 and such that the internal transitions
allow re-interpretion as tensor-product spline patches with two internal double
knots. Related, but structurally different, in the polar patch layout collapsed















bi-3 patches allow for a simple C1 surface construction also for high valences
[MKP07].


2 G1 continuity

Consider n parameterically C1 patches

b" : D ]R2 R3, k =,...,n (1)

meeting at a central point b (0, 0) p such that b (u, 0) b 1(0, u). We do
not (yet) assume that D is the unit square but just that the origin as a corner
with two independent edge directions emanating from it. We assume the patches
are not singular at the origin in the sense that 02b (0, 0) x d1b (0, 0) / 0 where
aO denotes differentiation with respect to the {th argument.



b

2b 1 b(u, 0) b 1(0, u)
U 1


Figure 1: Indexing and parameterization of adjacent patches at a vertex of
valence n. Here b =1 b" if k 1.

To make the n patches form a C1 surface, we want to enforce logically
symmetric (unbiased) G1 constraints.

Definition 1 (Unbiased G1 constraints) With ak a sufficiently smooth, uni-
variate scalar-valued function, the unbiased G1 constraints between consecutive
patches are
02bk(u, 0) + d1b (0, u) = k(u)ib k(u, 0), (2)
If a 0, the constraints enforce parametric C1 continuity. We abbreviate the
fth derivative of a evaluated at 0 as a .

Justification We can derive lower bound statements from this special unbiased
setting since an algorithm must work in general, i.e. for all data, and therefore
also for input whose symmetries suggest an unbiased treatment, e.g. no depen-
dence on the indexing or ordering of patches and no bias based on the geometric
interpretation.
We now add the assumption that each bk is twice continuously differentiable
at (0, 0) (as are the polynomial pieces of a spline patch).















Definition 2 (Spline) A C" spline patch is a patch b that is internally pa-
rameterically C', in particular across finitely many (domain knot) lines. At
any boundary knot (intersection of a knot line with the boundary of the do-
main), on either side of the knot line, 8d0dbk) is well-defined for i+ j < s + 1.
And the patch is regular in the corners: d2bW(0, 0) x lb (0, 0) / 0.

We can then differentiate along (the respective domain edge of) the common
boundary bW(u, 0) b b-(0, u):

(dD2b )(u, 0) + (d2dlb 1)(0, u)= a(u,. hk(u, 0) +(a)'(u)1bk (u, 0). (3)

When we evaluate at u = 0 then

at (0, 0), d012b + 02d1b = a D bk + a dibk. (4)

If n is even then the alternating sum of the left hand sides vanishes,
n
at (0,0), (-_l)k(002bk +0281bk 1) 0 (5)
k=1
and therefore so must the right hand side
n n
at (0,0), 0 -1) a b + (-1)k a b. (6)
k=1 k=1
In particular, if the patches join smoothly and therefore have a unique normal
n E R3 at p then with denoting scalar product,

if n is even, at (0,0) 0 (-1)k a n. b k. (7)
k=l
This is the vertex-enclosure constraint (see e.g. [Pet02, 1' -'" j).
We now focus on the case n = 4.

Definition 3 (tangent X) Ifn 4, dlb1(0,0) -dib3(0,0) and db2(0,0)
-d1b4(0,0) then the tangents form an X.

Lemma 1 (tangent X) If the tangents form an X, then al a3 and a2 a4.


Proof If the tangents form an X then n = 4 and ,', = k = 1, 2, 3, 4 so that
(6) simplifies to

at (0,0), 0 = (a a3)ib1 (a a4)91b2. (8)

Since the patches are not degenerate, both summands have to vanish, implying
the claim. III
We now consider the unbiased G1 transition between two C1 spline patches.
We focus on an interior boundary vertex (corresponding to an interior knot on















the common boundary). That is, we consider a point where four polynomial
pieces meet such that b1 and b2 belong to one spline patch and b3 and b4 are
adjacent pieces of the edge-adjacent spline patch (Figure 2). Since each spline
patch is internally parameterically C1,


a2 0 4.


2
b2 bl
1


Figure 2: Interior knot join on the boundary (solid) between two splines. The
first spline has polynomial pieces b1 and b2.


Lemma 2 (C1 spline, interior knot) Let (0, 0) be the parameter associated
with an interior knot on the boundary common to two C1 splines joined by
unbiased G1 constraints. Then


1 3
ao = -ao,


a(b1 -2 b3) + (a a3)tl, tk := db(0, 0).
0 1 1 1 tk : 01


Proof The parametric C1 constraints imply 81b1(0, 0)
n = 4, (10) follows. By (9), (6) specializes to


-1b3 (0, 0) and since


at (0, 0), 0 a 2b + a 1b3 + a'1bl + a3 b3
Sa W(b1 12b3) + (a' a3)tl


as claimed. I||
So, remarkably, when two spline patches meet along a common boundary, unbi-
ased G1 constraints across this boundary imply the constraint (11) exclusively
in terms of derivatives along the boundary.

Lemma 3 (C2 spline, interior knot) Let (0, 0) be the parameter associated
with an interior knot on the boundary common to two C2 splines joined by
unbiased G1 constraints. Then, in addition to (10),

a l = a3, (13)
at (0,0) : 0= a(b b3) + 4a 2bl + (a a)t. (14)


at (0,0) : 0















Proof Since the splines are C2, Ob1l(0, 0) = b3(0, 0). Then (6) implies (13).
By assumption, for the spline-internal boundary,

for k =2,4, at (0,0), 020102b + 12W1bk = 0. (15)

Differentiating (3) once more along the (direction corresponding to the) common
boundary, we obtain for

= 1, 3, at (0, 0), 9119t2bk + 92291b = a + 2a' b + a" lb.
(16)
Summing the two instances of (16) and subtracting the two instances of (15)
eliminates the mixed derivatives and yields at (0, 0)

0 1 a3b1 + 2a{ b1 + a ib1 + a ,b3 + 2a b3 + a 9b3. (17)
1alol a ag a 2a O~b3 aOl" (17)
Then C2 continuity implies (14). |||

2.1 Vertex-localized construction and linear a
Equation (4) shows that the Taylor expansions up to order two of the patches
joining at a point are strongly intermeshed in the sense that they constrain a
common set of derivatives. So, to avoid having to solve a global system, we
define the following.

Definition 4 (vertex-localized construction) A construction is G1 vertex-
localized if we can solve the unbiased G1 constraints (2) and (3) on the second-
order Taylor expansion D1b"k, 0 < i,j, i + j < 2 at (0, 0) corresponding to a
vertex, independent of the solutions at its neighbors.

Note that we allow the local solutions to depend on the a priori known
local connectivity and the valence of the neighbors in particular but not on the
local patch expansion computed there. In the most challenging case, the vertex
enclosure constraint (7) applies at each vertex. Therefore a vertex-localized
construction implies that the second-order Taylor expansion is set independently
at each vertex of the quad mesh. It is to this scenario and the unrestricted choice
of geometric instantiation that we take into account, when we prefix a statement
with in general.
Along a boundary, we index the th derivative of the scalar map in the G1
constraints of the jth spline piece meeting, as i.

Lemma 4 (piecewise linear a) In general, a vertex-localized construction of
unbiased G1 transitions between C1 spline patches with everywhere at most
linear a is not possible.

Proof Consider a vertex surrounded by vertices of valence n = 4. Then vertex-
localized construction implies that a0,o 0. Assume now that immediate neigh-
bor interior knots exist. Then local construction implies also a, 1 0 (and
a0,1 0) for these interior boundary vertexs. Shifting the focus to one such

























Figure 3: Propagation of ,, i = 0 in Lemma 4.


interior boundary vertex, say the one corresponding to ao,_1 we observe
that its tangents form an X since the two splines, one at either side, are C1.
So Lemma 1 implies ao, 2 0 since ao,o 0 and again this interior boundary
vertex's tangents form an X. In this manner, X formations and ao,j 0 prop-
agate (see the arrows in Figure 3 for illustration). Once the propagation meets
an original vertex with valence n j 4 (whether or not we have interior knots),
local construction clashes with Lemma 1. 111
Unbiased G1 constraints imply a local, unbiased choice of the tangent directions,
namely such that (see [Pet94, Prop 3])
2i7
a (0) : 2cos-. (18)
n
Corollary 1 If n 4, a local, unbiased choice of the tangent directions and a
linear imply n = nk+2

Proof The claim follows from Lemma 1 since by the unbiased choice a (0) := 0
and a (1) : 2cos2 ||
Lastly, we characterize a source of poor shape in smooth surface construc-
tions [Pet01b].

Lemma 5 (Flatness at saddle points) A curve segment c of an unbiased
construction, emanating from a higher-order saddle point p := c(0) and such
that
c' := et, : R R3, deg(e) < 1, 7 : R- R, deg(7) < 1, (19)
is planar. If the saddle is symmetric, c is a straight line segment.

Proof Let n be the normal at p and, without loss of generality, 7(u) := 1 +i-y
for some ji E R. Then c'(0) (0), c"(0) '(0)+e(0)y and c"'(0) 2f'(0)ii.
At a higher-order saddle point, the normal curvature is zero, and therefore
nc"(0) 0. This implies nf'(0) = 0 and nc"'(0) 0 If the saddle is symmetric
then c'(0) and c"(0) are collinear and so is c"'(0) 2'(0)71. I||
A higher-order saddle points, such as the center of the monkey saddle, should
have non-zero Gauss curvature apart from the point. Therefore, we will later
disqualify constructions that force straight boundary segments for such non-flat
geometry.















3 Lower bounds for degree bi-3


We now argue that, in general, vertex-localized construction with splines of
degree bi-3 (bicubic) is possible only if the spline patches have at least two
internal knots per edge and at least one that is a double knot.
Since we specialize to polynomial b of polynomial degree 3, a simple alge-
braic argument [Pet91, Cor. 2.2] shows that a is a rational function, a -:
with numerator of polynomial degree deg(/) < 6 and deg(y) < deg(/) 1 and
such that 3 and y have no factor in common. In fact, for unbiased constructions
we can conclude a lower bound on the degree.

Lemma 6 (a restricted) If the two patches bW and b 1 satisfying an unbi-
ased G1 constraint are of degree bi-3 and the boundary segment b (u, O) should
not be forced to be a line segment then

a -, (deg(3), deg(-)) {(2,1), (2, 0), (1, 1), (1, 0), (0, 0)} and (20)

dib"(u,O) = (u)7(u), deg(=) 2 -deg(7). (21)



Proof We may assume that 3 and 7 are relatively coprime. Then the
unbiased G1 constraint (2) implies that 7 is a (scalar) factor of dibk(u,0) E
R3, the (vector-valued) derivative of the boundary. By the assumptions, 0 <
deg(0ib'(u,0)) < 2. Since deg(2b'(u, 0) + d1b -l(0, u)) < 3 this limits
deg(3) < 2 and deg(Q) < 1. 1||
Scaling numerator and denominator, we may assume that y(u) := 1 + "yu.

Corollary 2 (polynomial vertex a) For a boundary segment of degree 3 di-
rectly attached to a vertex, a is linear.

Proof Choosing (deg(3),deg(7)) E {(2,1), (2,0), (1, 1)} forces the degree 3
curve segment to be of the form (19). Since we require in general more flexibil-
ity than forced straight line segments, Lemma 5 shows that a must be linear or
constant for segments emanating from the (original) vertices. III
Now recall that vertex-localized construction in general implies that position,
first and second derivative are specified at each endpoint of a (piecewise) G1 join
between splines.

3.1 No internal knot
Without internal knot each bW is a Bezier patch and its boundary curve a single
polynomial piece of degree 3. The second order expansion at a point p interferes
with the expansion at its neighbor. In particular, for valence n = 2m > 4,
Equation (7) links the second order Taylor expansions of all direct neighbors,
preventing a vertex-local construction.















3.2 1-fold internal knots only
Since the knots are of multiplicity 1 and the degree is bi-3, we have two C2 spline
patches meeting along a common boundary. Then each additional interior knot
corresponds to one additional boundary curve segment of degree 3. However,
CO, C1 and C2 constraints plus (14) of Lemma 3 impose four vector-valued
constraints. This is the intuition underlying the next lemma.

Lemma 7 (1-fold knots only) In general, no vertex-localized construction en-
forcing unbiased G1 constraints is possible using only C2 splines of degree bi-3.

Proof For the first and the last segment of the piecewise cubic spline bound-
ary curve, position, first and second derivative are given by vertex-local con-
struction. It is easy to check that if the boundary curve has two segments, there
are not enough degrees of freedom to, in general, enforce C2 continuity, but for
three segments C2 continuity uniquely determines all segments. However, this
construction leaves (14) unresolved at the two interior knots and therefore, in
general, yields no solution to (2). Inserting an additional knot adds another
curve segment. Let segment b3(u, 0) be given and b(u, 0) the result of adding
a knot (the segments are arranged as in Figure 2). By Lemma 4, not all a can
be linear. (Anyhow, if a is linear, it is fixed by (10) and (13) and therefore the
free (B-spline) control point must be used to resolve (14). That is, we do not
gain degrees of freedom to enforce (2).)
We therefore assume that (deg(3), deg(7)) E {(2, 1), (2, 0), (1,1)}. Then i1b (u, 0) :
(u)7(u), a linear vector-valued polynomial times the scalar (possibly constant)
factor 7(u) := 1+7iu. By (10) and (13) and the C2 constraints, constraint (14)
becomes at (0, 0)

0 a( b3 2 lf'(0)) + 4a 0b3 + (a a')ts. (22)
=:v

By C1 continuity (0)7(0) -= (0) -ts and hence the C2 constraint bb3
(0)7i + '(0) -t37i + '(0) implies

'(0) =ts1 + 12b3(0, 0). (23)

Therefore, at (0,0), v = db3 2y7(t371 + b3). Since, in general, the scalar
71 can not force v 0 a = a3 and = 0 = al must hold in order for (22) to
hold.
Now consider a vertex of valence n / 4 so that the local, unbiased choice
of the tangent directions (18) implies a / 0 at the vertex. By Corollary 2,
the segment emanating from a vertex has deg(a) < 1. Consider the segment
bl(u,0) with a1 not linear closest to the vertex in the sense that its neighbor
segment has a3 linear. Then a3 0 implies a3 = 0. Applying (10)
and (13) repeatedly to the linear a.,j when approaching the vertex, propagates
a.j = 0. This is incompatible with a / 0 at the vertex, completing the proof.
III















3.3 One double internal knot
Due to the vertex-local construction and since the double (not triple) knot im-
plies C1 continuity, the two-piece cubic boundary curve common to the two
spline patches is completely determined. Moreover, the values of a are de-
termined at the endpoints of the curve by the unbiased choice of the tangent
directions (18), page 7.

Lemma 8 (one double interior knot) In general, no vertex-localized con-
struction enforcing unbiased G1 constraints is possible, using bi-3 splines with
one internal double knot.

Proof let (0, 0) corresponding to the breakpoint of the boundary curve. Here
Lemma 2 applies and all vectors in (11) are fixed. By Corollary 2, a1 and a3
must both be linear. The claim follows since, in general, the only remaining
single free scalar a1(0) =: a = -a_3(0) cannot always enforce the vector-valued
constraint (11). |||


4 Discussion and Conclusion

Using the Justification (page 3) and the assumption that all constructions are
vertex-local, we could derive lower bounds from unbiased, logically symmetric
constructions. Conversely, the construction in [FP08] establishes a tight upper
bound on the number of polynomial pieces: It uses double internal knots and
does not have the shape problem characterized by Lemma 5.
It has long been know that if all valences are odd or tangents are in an X
configuration, vertex-enclosure does not impose constraints and simple Bezier
constructions are possible (e.g. [Pet91, GZ94]). More global constructions, sin-
gular parameterization, or restrictions on the valence, for example by split-
ting patches, can allow for simpler constructions, e.g. ;. '1".], [PBP02, 9.11],
[Pet91, Pet95b]. We note also that ,." n1 then we can choose linear a1 and
a3 with al = a^ and a 0 to enforce (11) and arrive at a 2x2-split construction
that is ruled out in general by Lemma 8.
Lemma 4 prevents a vertex-local solution with all .... linear. When the
lemma is specialized by fixing the degree to be 3, increasing the patch continuity
to C2 and choosing .... := la00 + o0q then we arrive at the statement (but
not the proof sketch) of [SWWL04, Thm 3.1]. So, if we insist on single internal
knots everywhere, a subdivision-like construction as proposed in [SWWL04] is
justified by Lemma 4.
The results provide a checklist for constructions that are otherwise difficult
to verify. Lemma 8 for example indicates that there must be a subtle error in the
construction [HBCi -] although, as mentioned above, such 2x2-split construction
can succeed in special cases, such as smoothing a cube.
Constructions of smooth surfaces with one patch per quad are shown possible
for degree bi-5, for example by [MYP08]. For degree bi-4, a single knot (a 2x2-
split) must be introduced (see e.g. [Pet95a]).















Remarkably, the results in Section 2 do not depend on the degree or even
the polynomial nature of splines, but assume only sufficiently smooth functions,
that are piecewise with smooth transitions between the pieces.


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