Exam Toric degenerations of Bezier patches - TORIC DEGENERATIONS OF BEZIER PATCHES arXiv:1006.4903v2[cs.GR 30 Dec 2010 LUIS DAVID GARC IA-PUENTE FRANK

Exam Toric degenerations of Bezier patches - TORIC...

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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 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
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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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  • Fall '14
  • LuisDavidGarcia-Puente
  • Statistics, Convex hull, Polytope, Convex combination, Convex hull algorithms

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