Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 11 pps

... 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...

Ngày tải lên: 10/08/2014, 02:20

... 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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... algorithm. 3-5 4 0-3 325 8-8 Springer Berlin Heidelberg New York Monique Teillaud Jean-Daniel Boissonnat INRIA Sophia-Antipolis 2004 route des Lucioles B.P. 93 06902 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...

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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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... 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 tangency between C 1 and C 2 oc- curs when  (x 1 − ... divide curves to form x-monotone sub -curves) . • If an isolated vertex has the same selection mark as its surrounding face, then this vertex is redundant and is removed from...

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... 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+ε ) for any ε>0. This ... 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) diagram...

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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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... vertical parts explicitly.) For simplicity of exposition, we will also assume that the curve or surface is bounded. The algorithm can also deal with curves and surfaces that 5 Meshing of Surfaces 211 Among ... (k − 1)-ball or in the empty set, for every k ≥ 0. (For example, the non-empty intersection of a 2-dimensional Voronoi face F with the surface must be a curve segment.) More...

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... 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 from one plane ... 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 5.23...

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