Triangulations of Line Segment Sets in the Plane

β-skeletons are well-known neighborhood graphs for a set of points. We extend this notion to sets of line segments in the Euclidean plane and present algorithms computing such skeletons for the entire range of β values. The main reason of such extension is the possibility to study β-skeletons for points moving along given line segments. We show that relations between β-skeletons for β > 1, 1-skeleton (Gabriel Graph), and the Delaunay triangulation for sets of points hold also for sets of segments. We present algorithms for computing circle-based and lune-based β-skeletons. We describe an algorithm that for β ≥ 1 computes the β-skeleton for a set S of n segments in the Euclidean plane in O(n 2 α(n) log n) time in the circle-based case and in O(n 2 λ4(n)) in the lune-based one, where the construction relies on the Delaunay triangulation for S, α is a functional inverse of Ackermann function and λ4(n) denotes the maximum possible length of a (n, 4) Davenport-Schinzel sequence. When 0 < β < 1, the β-skeleton can be constructed in a O(n 3 λ4(n)) time. In the special case of β = 1, which is a generalization of Gabriel Graph, the construction can be carried out in a O(n log n) time.

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“…Chew and Kedem [4] defined the Delaunay triangulation for line segments. Their definition was generalized by Brévilliers et al [2].…”

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confidence: 99%

$$\beta $$-skeletons for a Set of Line Segments in $$R^2 $$

Kowaluk

Majewska

2015

Fundamentals of Computation Theory

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“…In this section, we recall the main results about segment triangulations given in [4]. They generalize the concept of triangulation to a set of disjoint segments in the plane.…”

Section: Segment Triangulationsmentioning

confidence: 99%

“…It has been shown that the segment Delaunay triangulation exists for any set S, is unique if S is in general position, and is dual to the segment Voronoi diagram. As for point sets, the legal edge property has been defined for segment triangulations in [4]. A more intuitive formulation is: This led to a local characterization of the segment Delaunay triangulation:…”

Section: Definition 2 a Segment Triangulation Of S Is Delaunay If Thmentioning

confidence: 99%

“…In this work we address the question of flip algorithm for the segment triangulations that have been introduced in [4]. Given a finite set S of disjoint line segments in the plane, a segment triangulation of S is a maximal set of disjoint triangles, each of them having its vertices on three distinct sites of S (see Figure 1).…”

Section: Introductionmentioning

confidence: 99%

“…A topological dual of the segment Voronoi diagram has also been used to implement efficiently the construction of the segment Voronoi diagram in the CGAL Library [15]. In [4], we have given a local characterization of the segment Delaunay triangulation among the family of all segment triangulations of S as well as a local characterization of its topology.…”

Section: Introductionmentioning

confidence: 99%

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Flip Algorithm for Segment Triangulations

Brevilliers

Chevallier

Schmitt

Lecture Notes in Computer Science

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Abstract. Given a set S of disjoint line segments in the plane, which we call sites, a segment triangulation of S is a partition of the convex hull of S into sites, edges, and faces. The set of faces is a maximal set of disjoint triangles such that the vertices of each triangle are on three distinct sites. The segment Delaunay triangulation of S is the segment triangulation of S whose faces are inscribable in circles whose interiors do not intersect S. It is dual to the segment Voronoi diagram. The aim of this paper is to show that any given segment triangulation can be transformed by a finite sequence of local improvements in a segment triangulation that has the same topology as the segment Delaunay triangulation. The main difference with the classical flip algorithm for point set triangulations is that local improvements have to be computed on non convex regions. We overcome this difficulty by using locally convex functions.

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Constant Factor Approximation for Intersecting Line Segments with Disks

Kobylkin

2018

Lecture Notes in Computer Science

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Triangulations of Line Segment Sets in the Plane

Cited by 7 publications

References 14 publications

$$\beta $$-skeletons for a Set of Line Segments in $$R^2 $$

$$\beta $$-skeletons for a Set of Line Segments in $$R^2 $$

Flip Algorithm for Segment Triangulations

Constant Factor Approximation for Intersecting Line Segments with Disks

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