By Juncong Lin, Xiaogang Jin, Zhengwen Fan, Charlie C. L. Wang (auth.), Falai Chen, Bert Jüttler (eds.)
This publication constitutes the refereed complaints of the fifth overseas convention on Geometric Modeling and Processing, GMP 2008, held in Hangzhou, China, in April 2008.
The 34 revised complete papers and 17 revised brief papers awarded have been conscientiously reviewed and chosen from a complete of 113 submissions. The papers conceal a large spectrum within the quarter of geometric modeling and processing and tackle themes equivalent to curves and surfaces, electronic geometry processing, geometric characteristic modeling and popularity, geometric constraint fixing, geometric optimization, multiresolution modeling, and purposes in laptop imaginative and prescient, snapshot processing, clinical visualization, robotics and opposite engineering.
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Additional resources for Advances in Geometric Modeling and Processing: 5th International Conference, GMP 2008, Hangzhou, China, April 23-25, 2008. Proceedings
Bounding the Distance between a Loop Subdivision Surface 41 It is obvious that B(0, 0) = B(1, 0) = B(0, 1) = 0, and B(v, w) is a piecewise quartic triangular B´ezier function over Ω away from (0, 0). Let β(n) = max(v,w)∈Ω B(v, w), we have the following theorem on the maximal distance between S(v, w) and F(v, w): Theorem 3. The distance between an extraordinary Loop patch S of valence n and the corresponding limit face F is bounded by max (v,w)∈Ω S(v, w) − F(v, w) ≤ β(n)M , (11) where β(n) is a constant that depends only on n, the valence of S.
44 Z. Huang and G. Wang 6 Comparison Both a control mesh and its corresponding limit mesh can be employed to approximate a Loop surface in practical applications. This section compares these two approximation representations within the framework of the second order diﬀerence techniques. The distance between a Loop patch S of valence n and its control mesh can be bounded in terms of the second order norm M as : max (v,w)∈Ω S(v, w) − F(v, w) ≤ Cλ (n)M, where Cλ (n) = β(n) λ−1 i=0 ri (n) 1 − rλ (n) λ≥1 , (15) .
N are taken modulo n. Let ui ∈ [0, 1] be the parameter corresponding to the curve between v and vi , the surface patch S i is then parameterized as illustrated in Figure 1. e. G1 continuous. This means that the surface is C∞ everywhere except at the inner patch boundaries where it has continuously varying tangent planes. Let S i and S i−1 be two adjacent tensor product B´ezier patches parameterized as in Figure 1. S i and S i−1 join at the common boundary with tangent plane continuity, denoted G1 , if and only if there exist three scalar functions Φi , νi and μi such that Φi (ui ) ∂S i ∂S i ∂S i−1 (ui , 0) = νi (ui ) (ui , 0) + μi (ui ) (0, ui ) , ∂ui ∂ui+1 ∂ui−1 (2) where νi (ui ) · μi (ui ) > 0 (preservation of orientation, avoiding ridges) and ∂S i ∂S i ∂ui (ui , 0) × ∂ui+1 (ui , 0) = 0 (well deﬁned normal vectors).
Advances in Geometric Modeling and Processing: 5th International Conference, GMP 2008, Hangzhou, China, April 23-25, 2008. Proceedings by Juncong Lin, Xiaogang Jin, Zhengwen Fan, Charlie C. L. Wang (auth.), Falai Chen, Bert Jüttler (eds.)