arXiv:1006.4903v2
[cs.GR]
30 Dec 2010
TORIC DEGENERATIONS OF B
´
EZIER PATCHES
LUIS DAVID GARC
´
IA-PUENTE, FRANK SOTTILE, AND CHUNGANG ZHU
Abstract.
The control polygon of a B´
ezier curve is well-defined and has geometric
significance—there is a sequence of weights under which the limiting position of the curve
is the control polygon. For a B´
ezier surface patch, there are many possible polyhedral con-
trol structures, and none are canonical. We propose a not necessarily polyhedral control
structure for surface patches, regular control surfaces, which are certain
C
0
spline surfaces.
While not unique, regular control surfaces are exactly the possible limiting positions of a
B´
ezier patch when the weights are allowed to vary.
1.
Introduction
In geometric modeling of curves and surfaces, the overall shape of an individual patch is
intuitively governed by the placement of control points, and a patch may be finely tuned
by altering the weights of the basis functions—large weights pull the patch towards the
corresponding control points. The control points also have a global meaning as the patch
lies within the convex hull of the control points, for any choice of weights.
This convex hull is often indicated by drawing some edges between the control points.
The rational bicubic tensor product patches in Figure 1 have the same weights but different
control points, and the same 3
×
3 quadrilateral grid of edges drawn between the control
points. Unlike the control points or their convex hulls, there is no canonical choice of these
Figure 1.
Two rational bicubic patches.
Key words and phrases.
control polytope, B´
ezier patch.
Work of Garc´
ıa-Puente and Sottile supported in part by the Texas Advanced Research Program under
Grant No. 010366-0054-2007.
Research of Sottile supported in part by NSF grant DMS-070105.
Research of Zhu supported in part by NSFC grant 10801024, U0935004, and this work was conducted
in part at Texas A&M University.
1

2
L. GARC
´
IA-PUENTE, F. SOTTILE, AND C-G. ZHU
edges. We paraphrase a question posed to us by Carl de Boor and Ron Goldman: What
is the significance for modeling of such control structures (control points plus edges)?
We provide an answer to this question. These control structures, the triangles, quadri-
laterals, and other shapes implied by these edges, encode limiting positions of the patch
when the weights assume extreme values.
By Theorems 5.1 and 5.2, the only possible
limiting positions of a patch are the control structures arising from
regular decompositions
(see Section 4) of the points indexing its basis functions and control points, and any such
regular control structure is the limiting position of some sequence of patches.
Here are rational bicubic patches with the control points of Figure 1 and extreme weights.

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