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

... S.ABCDEFGABCDEFG0 100 2000 100 100 100 0 100 200 100 100 0200 100 100 100 ABCDEFGABCDEFG0 100 2000 100 100 100 0 100 200 100 − 100 0200 100 100 100 100 0 100 100 0fg0Fig. ... isotopic to the cross-section. The isotopy hasonly 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 ... theso-called α-complex and α-shape of a collection of balls.α-complex and α-shape.Given a sample P , consider the collection of balls of square radius α centeredat these points1. For each...

Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 2 pot

... 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 Arrangement2 <Traits,Dcel>12class-template. It represents theplanar embedding of a set of x-monotone planar curves that are pairwise dis-joint in their ... recent Ecg project, the Max-Planck Institute for Computer Sci-ence in Saarbr¨ucken, Germany, initiated the project Exacus —Eﬃcient andExact Algorithms for Curves and Surfaces [4]— where we develop...

Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 6 pot

... 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 andβ 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 beingtreated ... ﬁrstthat q(c) > 0, q(b) < 0. Then if v = 1, by Sturm’s theorem q(α) > 0andα<β.Ifv = −1, q(α) < 0andα>β.Ifv = 0, then q(α)=0andα = β.If now q(c) < 0, q(b) > 0, we negate...

Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 1 docx

... algorithmicfoundations for eﬀective computational geometry for curves and surfaces. Thisbook covers two main approaches.In a ﬁrst part, we discuss exact geometric algorithms for curves and sur-faces. We ... 223EditorsABC Effective Computational Jean-Daniel Boissonnat Monique Teillaud With 120 Figures and 1 Table Geometry for Curves and Surfaces 6 E. Fogel, D. Halperin, L. Kettner, M. Teillaud, R. ... YorkMonique Teillaud Jean-Daniel Boissonnat INRIA Sophia-Antipolis2004 route des LuciolesB.P. 9306902 Sophia-Antipolis, FranceE-mail: Jean-Daniel .Boissonnat@ sophia.inria.frMonique .Teillaud@ sophia.inria.frPrefaceComputational...

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

... singlepoint. We transform the requirement that thepredicates will evaluate to suﬃciently-far-from-zero values into a geometric distance require-ment.An outer tangency between C1and C2oc-curs when(x1− ... divide curves to form x-monotone sub -curves) .• If an isolated vertex has the same selection mark as its surroundingface, then this vertex is redundant and is removed from the arrange-ment.Clearly, ... from two-dimensional arrangements of curves to three-dimensionalarrangements of surfaces is a major endeavor. In this section we report onprogress in and plans for coping with three-dimensional...

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

... O(nd+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 randomizedexpected time O(nd+ε) for any ε>0.This ... quadratic hypersurfaces. These dia-grams can be computed through linearization, a technique to be described infull generality in Sect. 2.5. Apollonius (or Johnson-Mehl) diagrams, althoughsemi-algebraic ... in text books on Computational Geometry [ 110, 67]. This chapter does not consider Voronoidiagrams deﬁned in more general spaces. Voronoi diagrams can be deﬁned inhyperbolic geometry without...

Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 5 pps

... sufficiently far apart, is close to a medial sphereof S (for the Hausdorff distance). We have therefore bounded the one-sidedHausdorff distance from the λ-medial axis (Voronoi diagram) of an ε-sampleP ... that on each VI, for all i<j, δi(x) <δj(x)iffx ∈ biijandδi(x)=δj(x)iffx ∈ bij. The induction follows.94 J-D. Boissonnat, C. Wormser, M. YvinecLemma 3. For any distinct i, ... as the zero-set of some real-valuedfunction over Rd, called a bisector-function in the following. Let us denote byB the set of bisector-functions. By convention, for any bisector-function...

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

... sjgreatest points for the lexicographic order withx>y>z,amongLi−∪l>jVl.• For j<j0, Vjis the set of sjsmallest points for the lexicographic order,among Li−∪l<jVl.• ... values:δ0<σ1<µ1< ···<σl<δ1, where µi:=σi+σi+12 for i =0, ,l−1, andδ0,δ1are any value such that ]δ0,δ1[ contains Σ. We denote byα0< ···<αmthis ... given for a more general classof curves, (that is, the class of curves for which (4.2) is ﬁnite - the rectiﬁable curves- ). Remark that (4.1) and (4.2) imply immediately a convergence theo-rem:...

Effective Computational Geometry for Curves & Surfaces - Boissonnat & Teillaud Part 9 pptx

... (k − 1)-ball or in the empty set, for every k ≥ 0. (For example, the non-empty intersection of a 2-dimensionalVoronoi 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 can ... ﬁrst formulated for ε-samples (with the same bound of0.1) by Amenta and Bern [22], see also Theorem 6 in Chap. 6 (p. 248) for arelated theorem. The extension to weak ε-samples is due to Boissonnat...

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

... axisε-sample.Amenta and Bern [23, 22] introduced a non-uniform measure of samplingdensity using the local feature size. For ε>0 a sample P of a surface S iscalled an ε-sample of S if every point x on S has ... This result is false ingeneral for non smooth curves, as illustrated in Fig. 6.14. It is also false ingeneral for dense samples of smooth surfaces. In fact for almost any pointx ∈ R3\S, ... only works for smoothclosed surfaces and suﬃciently dense ε-samples. Main problem is the ﬁnal stepof the algorithm that extracts a surfaces from the set of candidate triangles. For practical...

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