Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 10 pot

... S. A B C D E F G A B C D E F G 0 100 200 0 100 100 100 0 100 200 100 100 0 200 100 100 100 A B C D E F G A B C D E F G 0 100 200 0 100 100 100 0 100 200 100 − 100 0 200 100 100 100 100 0 100 100 0 f g 0 Fig. ... 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 variet...

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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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... 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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... algorithmic foundations for effective computational geometry for curves and surfaces. This book covers two main approaches. In a first part, we discuss exact geometric algorithms for curves and sur- faces. We ... 223 Editors ABC Effective Computational Jean-Daniel Boissonnat Monique Teillaud With 120 Figures and 1 Table Geometry for Curves and Surfaces 6 E. Fogel...

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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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... 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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... (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.) Moreover, the inter- section ... 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- ever. It...

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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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