Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 3 docx

... set of roots of the polynomial 40 833 2484x 4 + 51 939 673y 4 − 664779204x 3 y − 24101506y 3 x +564185724x 2 y 2 − 250019406x 3 + 17767644y 3 +221120964x 2 y − 1 230 26916y 2 x + 16691919x 2 + 4764152y 2 +14441004xy ... [85, 196, 31 2]. For simplicity of exposition we describe the three-dimensional variant for arrangements of triangles. Let T = {t 1 ,t 2 , ,t n } be a collection...

Ngày tải lên: 10/08/2014, 02:20

... unit 3- sphere S3 – further details can be found in [1, Chap. III, Section 8.5]. This picture was obtained from a 3D model generated with the CGAL surface meshing algorithm. 3- 5 4 0 -3 32 5 8-8 Springer ... 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.i...

Ngày tải lên: 10/08/2014, 02:20

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

Ngày tải lên: 10/08/2014, 02:20

... triangulation of P. 2 .3. 3 Power Diagrams A construction similar to what we did for the Euclidean Voronoi diagrams of points and their dual Delaunay triangulations can be done for the so-called power ... 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. Wormser, M. Yvinec Fig. 2 .3. The poly...

Ngày tải lên: 10/08/2014, 02:20

... 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, ... 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-...

Ngày tải lên: 10/08/2014, 02:20

... 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...

Ngày tải lên: 10/08/2014, 02:20

... 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

... (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...

Ngày tải lên: 10/08/2014, 02:20

... 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 is a two-dimensional ... 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...

Ngày tải lên: 10/08/2014, 02:20

... 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 ∈ R 3 \S, there exists ... A 3 1 and A 3 contacts, and the points to A 4 1 and A 3 A 1 contacts. Moreover, one has the following incidences. At an A 4 1 point, six A 1 2 sheets and four A 3 1 curves meet. A...

Ngày tải lên: 10/08/2014, 02:20