Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 5 pps

... 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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... 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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... Algebraic Curves. Birkhäuser, Basel Boston Stuttgart, 1986. 68N30; 65D17; 57 Q 15; 57 R 05; 57 Q 55; 65D 05; 57 N 05; 57 N 65; 58 A 05; 68W 05; 68W20; Mathematics Subject Classification: 68U 05; 65D18; 14Q 05; 14Q10; ... algorithmic foundations for effective computational geometry for curves and surfaces. This book covers two main approaches. In a first part, we discuss exa...

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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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... successfully implemented for ar- rangements of triangles and of polyhedral surfaces. It computes the vertical 15 For a detailed discussion of what constitute well-behaved surfaces or surface patches ... polynomial 408332484x 4 + 51 939673y 4 − 664779204x 3 y − 2410 150 6y 3 x +56 41 857 24x 2 y 2 − 250 019406x 3 + 17767644y 3 +221120964x 2 y − 123026916y 2 x + 16691919x 2 + 4764 15...

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... quadratic hypersurfaces. These dia- grams can be computed through linearization, a technique to be described in full generality in Sect. 2 .5. Apollonius (or Johnson-Mehl) diagrams, although semi-algebraic ... 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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... 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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... works for surfaces with self- intersections, fold lines, or other singularities. On the other hand, it makes 1 http://www.cse.ohio-state.edu/˜tamaldey/surfremesh.html 5 Meshing of Surfaces 1 95 The ... structure that can be constructed easily. Fig. 5. 15 shows a mesh for a torus shape that is constructed in this way. Before meshing Fig. 5. 15. A triangular mesh for a torus con...

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... lower bound ψ? 3. The test (5. 5) for critical points in a silhouette involves second derivatives (cf. Exercise 15) . Is there a zero-dimensional system of equations for es- tablishing the topological ... D. Boissonnat, D. Cohen-Steiner, B. Mourrain, G. Rote, G. Vegter points. Unfortunately, this is not sufficient even for implicit curves in the plane. Indeed, the situation in figure 5...

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