Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 9 pptx

... connect them. For each point, we need to know how many parts of 5 Meshing of Surfaces 199 (b)(a) (c) ?? ⊕ ⊕ Fig. 5.10. (a) The ambiguous sign pattern can arise on a face of a cube. (b-c) The ambiguity ... other function ψ:Aψ-sample P ⊂ S for a function ψ: S → R + is a subset with the following property: For every point x ∈ S, there is a point p ∈ P , such that 198 J D. Boissonnat,...

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... York Monique Teillaud Jean-Daniel Boissonnat INRIA Sophia-Antipolis 2004 route des Lucioles B.P. 93 0 690 2 Sophia-Antipolis, France E-mail: Jean-Daniel .Boissonnat@ sophia.inria.fr Monique .Teillaud@ sophia.inria.fr Preface Computational ... algorithmic foundations for effective computational geometry for curves and surfaces. This book covers two main approaches. In a first...

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... independent part that implements the predicates and constructions for the sweep-line algorithm based upon the one-curve and two -curves analysis, as described in Sect. 1.3.1; see [48] for details. Most ... components. The main com- ponent is the Arrangement 2 <Traits,Dcel> 12 class-template. It represents the planar embedding of a set of x-monotone planar curves that are pairwi...

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... polynomial 408332484x 4 + 5 193 9673y 4 − 6647 792 04x 3 y − 24101506y 3 x +564185724x 2 y 2 − 2500 194 06x 3 + 17767644y 3 +22112 096 4x 2 y − 12302 691 6y 2 x + 16 691 919x 2 + 4764152y 2 +14441004xy + 1048 290 0x + 2305740y ... single point. We transform the requirement that the predicates will evaluate to sufficiently-far-from- zero values into a geometric distance require- ment. An outer t...

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... applications in biology, ecology, chem- istry and other fields (see Sect. 2 .9) . Euclidean Voronoi diagrams of non- punctual objects are also non-affine diagrams. They are of particular interest in robotics, ... O(n d+ε ) for any ε>0, where the constant of proportionality depends on ε, d and η. The vertices, edges and 2-faces of the diagram can be computed in randomized expected time O(n d...

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... that on each V I , for all i<j, δ i (x) <δ j (x)iffx ∈ b i ij and δ i (x)=δ j (x)iffx ∈ b ij . The induction follows. 94 J-D. Boissonnat, C. Wormser, M. Yvinec Lemma 3. For any distinct i, ... of the Apollonius di- agram of n circles in the plane has linear size. 98 J-D. Boissonnat, C. Wormser, M. Yvinec Theorem 14. Let B = {β ij } be a set of real-valued bisector-functions over R...

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... roots ∆ 1 > 0 ∧T>0 ∧ ∆ 2 > 0 {1, 1, 1, 1} ∆ 1 > 0 ∧(T ≤ 0 ∨ ∆ 2 ≤ 0) {} ∆ 1 < 0 {1, 1} ∆ 1 =0∧ T>0 {2, 1, 1} ∆ 1 =0∧ T<0 {2} ∆ 1 =0∧ T =0∧∆ 2 > 0 ∧R =0{2, 2} ∆ 1 =0∧ T =0∧∆ 2 > ... β =(q, ]c, d[), assuming for simplicity that α and β are simple roots of p and q.Ifb<c(resp. d<a)wehaveα<β(resp. β<α). Let us assume now that a<c<b<d(the other case...

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... s j greatest points for the lexicographic order with x>y>z,amongL i −∪ l>j V l . • For j<j 0 , V j is the set of s j smallest points for the lexicographic order, among L i −∪ l<j V l . • ... values: δ 0 <σ 1 <µ 1 < ···<σ l <δ 1 , where µ i := σ i +σ i+1 2 for i =0, ,l−1, and δ 0 ,δ 1 are any value such that ]δ 0 ,δ 1 [ contains Σ. We denote by α 0 < ··...

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... addition to those that are necessary for the curves in the two-dimensional vertical cross- sections: • We must be able find all slab points. 5 Meshing of Surfaces 2 29 For manifolds of codimension at ... isotopic to the cross-section. The isotopy has only deformed the curves vertically. Now, as we look at a slab from the top, the polar variety will form x- monotone non-crossing cur...

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... axis ε-sample. Amenta and Bern [23, 22] introduced a non-uniform measure of sampling density using the local feature size. For ε>0 a sample P of a surface S is called an ε-sample of S if every point x on S has ... This result is false in general for non smooth curves, as illustrated in Fig. 6.14. It is also false in general for dense samples of smooth surfaces. In fact for almost...

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