Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 4 pdf

... hypersurfaces (see Sect. 2 .4. 2). The related case of Apollonius (or Johnson-Mehl) diagrams is also described in Sect. 2 .4. The key to obtaining effective algorithms for computing those non-affine diagrams ... 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. Wo...

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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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... York Monique Teillaud Jean-Daniel Boissonnat INRIA Sophia-Antipolis 20 04 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 and surfaces. This book covers two main approaches. In a first...

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... and Exact Algorithms for Curves and Surfaces [4] — where we develop several C ++ software libraries for studying design and for experimenting with algorithms for arrangements of curves and surfaces. We ... 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...

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... polynomial 40 833 248 4x 4 + 51939673y 4 − 6 647 79204x 3 y − 241 01506y 3 x +5 641 85724x 2 y 2 − 25001 940 6x 3 + 17767 644 y 3 +221120964x 2 y − 123026916y 2 x + 16691919x 2 + 47 641 52y 2 + 144 41004xy + 1 048 2900x ... of algorithms for two-dimensional arrange- ments do not assume general position and handle degeneracies. 44 E. Fogel, D. Halperin, L. Kettner, M. Teillaud, R. We...

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... 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 distinct i, ... of the Apollonius di- agram of n circles in the plane has linear size. 98 J-D. Boissonnat, C. Wormser, M. Yvinec Theorem 14. Let B = {β ij } be a set of real-valued bisector-functions over R...

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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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... 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 can ... 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 B...

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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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... similar in nature to Boissonnat s early algorithm. 248 F. Cazals, J. Giesen Fig. 6. 14. For a non-smooth curve, some Voronoi centers may not converge to the medial axis ε-sample. Amenta and Bern ... 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 ∈...

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