Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 7 pdf

... 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 < ··...

Ngày tải lên: 10/08/2014, 02:20

... [ 276 , 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 ... r i ) 2 =(x − p i ) 2 (x d+1 + r j ) 2 ≤ (x −p j ) 2 for any j = i, and by subtracting both sides, it follows that Σ i (X i ) ≤ Σ j (X i ) for all j. 74 J-D. Boissonnat, C. Worm...

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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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... 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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... 51939 673 y 4 − 66 477 9204x 3 y − 24101506y 3 x +56418 572 4x 2 y 2 − 250019406x 3 + 177 676 44y 3 +221120964x 2 y − 123026916y 2 x + 16691919x 2 + 476 4152y 2 +14441004xy + 10482900x + 230 574 0y − 176 3465. ... 141y 2 +71 z 2 − 157xy +97xz − 111yz − 3x −6y − 17z − 7 . On the surface of a given quadric p, the intersection curves of p with the remaining quadrics induce a two-dimensiona...

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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 ... many text- books (see e.g. [ 67] ) recalls that state-of-the-art incremental constructions of Voronoi diagrams of points and power diagrams have an optimal randomized complexity. Theorem 17....

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... 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 cases being treated ... first that q(c) > 0, q(b) < 0. Then if v = 1, by Sturm’s theorem q(α) > 0and α<β.Ifv = −1, q(α) < 0andα>β.Ifv = 0, then q(α)=0andα = β. If now q(c) < 0, q(b) > 0, we...

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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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... the x- y-plane which does not lie on the projection of C, for example, at the “inter- mediate” vertical lines from the cross-sections (the open circles in Fig. 5.17b). It is now a straightforward ... 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...

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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...

Ngày tải lên: 10/08/2014, 02:20