Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 6 pot

... 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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... curve- type 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 ... 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 pairwise...

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... illustrated in Fig. 6. 6. The restricted Delaunay triangulation is also most convenient to introduce the so-called α-complex and α-shape of a collection of balls. α-complex and α-shape. Given a sample ... 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 curves f...

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... 223 Editors ABC Effective Computational Jean-Daniel Boissonnat Monique Teillaud With 120 Figures and 1 Table Geometry for Curves and Surfaces 6 E. Fogel, D. Halperin, L. Kettner, M. Teillaud, R. ... York Monique Teillaud Jean-Daniel Boissonnat INRIA Sophia-Antipolis 2004 route des Lucioles B.P. 93 069 02 Sophia-Antipolis, France E-mail: Jean-Daniel .Boissonnat@ sophia...

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... 5193 967 3y 4 − 66 4779204x 3 y − 24101506y 3 x + 564 185724x 2 y 2 − 250019406x 3 + 17 767 644y 3 +221120 964 x 2 y − 1230 269 16y 2 x + 166 91919x 2 + 4 764 152y 2 +14441004xy + 10482900x + 2305740y −1 763 465 . ... smooth molecular surfaces, 26 proposed by Richards [294] and imple- mented by Connolly [103], which can be derived from the union. For more 26 The so-called smooth mol...

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... [2 76, 37] and in text books on Computational Geometry [110, 67 ]. This chapter does not consider Voronoi diagrams defined in more general spaces. Voronoi diagrams can be defined in hyperbolic geometry ... following sections is to present effective algorithms for a variety of Voronoi diagrams for which some additional structure can be exhibited. 68 J-D. Boissonnat, C. Wormser, M. Yv...

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... sufficiently far apart, is close to a medial sphere of S (for the Hausdorff distance). We have therefore bounded the one-sided Hausdorff distance from the λ-medial axis (Voronoi diagram) of an ε-sample P ... 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 dist...

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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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... first formulated for ε-samples (with the same bound of 0.1) by Amenta and Bern [22], see also Theorem 6 in Chap. 6 (p. 248) for a related theorem. The extension to weak ε-samples is due to Boissonnat ... called a weak ε-sample (a weak ψ-sample, respectively). (Originally, this was called a loose ε-sample [64 ].) The difference between ε-samples and weak ε-samples is not too big, how-...

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... 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 any point x ∈ R 3 \S, ... intersect shallowly, see Fig. 6. 23 for an illustration. This result was already present in [30] for the two-dimensional case. 6 Delaunay Triangulation Based Surface Reconstruction 26...

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