Union and split operations on dynamic trapezoidal maps☆☆This work was partially supported by ESPRIT Basic Research Action r. 7141 (ALCOM II). A preliminary version appeared as [22].

AbstractWe propose algorithms to perform two new operations on an arrangement of line segments in the plane, represented by a trapezoidal map: the split of the map along a given vertical line D, and the union of two trapezoidal maps computed in two vertical slabs of the plane that are adjacent through a vertical line D. The data structure we use is a modified Influence Graph, still allowing dynamic insertions and deletions of line segments in the map. The algorithms for both operations run in O(sD logn+log2n) time, where n is the number of line segments in the map, and sD is the number of line segments intersected by D

Triangulating the Real Projective Plane

We consider the problem of computing a triangulation of the real projective plane P2, given a finite point set S={p1, p2,..., pn} as input. We prove that a triangulation of P2 always exists if at least six points in S are in general position, i.e., no three of them are collinear. We also design an algorithm for triangulating P2 if this necessary condition holds. As far as we know, this is the first computational result on the real projective plane

Implementing Delaunay Triangulations of the Bolza Surface

The CGAL library offers software packages to compute Delaunay triangulations of the (flat) torus of genus one in two and three dimensions. To the best of our knowledge, there is no available software for the simplest possible extension, i.e., the Bolza surface, a hyperbolic manifold homeomorphic to a torus of genus two. In this paper, we present an implementation based on the theoretical results and the incremental algorithm proposed last year at SoCG by Bogdanov, Teillaud, and Vegter. We describe the representation of the triangulation, we detail the different steps of the algorithm, we study predicates, and report experimental results

Generalizing CGAL Periodic Delaunay Triangulations

Even though Delaunay originally introduced his famous triangulations in the case of infinite point sets with translational periodicity, a software that computes such triangulations in the general case is not yet available, to the best of our knowledge. Combining and generalizing previous work, we present a practical algorithm for computing such triangulations. The algorithm has been implemented and experiments show that its performance is as good as the one of the CGAL package, which is restricted to cubic periodicity

Union and Split Operations on Dynamic Trapezoidal Maps

We propose here algorithms to perform two new operations on an arrangement of line segments in the plane, represented by a trapezoidal map: the split of the map along a given vertical line $D$, and the union of two trapezoidal maps computed in two vertical slabs of the plane that are adjacent through a vertical line $D$. The data structure is a modified Influence graph, still allowing dynamic insertions and deletions of line segments in themap. The algorithms for both operations run in $O(s_D \log n +\log^2 n)$ time, where $n$ is the number of line segments in the map, and $s_D$ is the number of line segments intersected by $D$

Towards dynamic randomized algorithms in computational geometry

Computational geometry aims to design and analyze algorithms for solving geometric problem. It is a recent field of theorical computer science, that rapidly developed since it appeared in M.I Shamos' thesis in 1978. Randomization allows to avoid the use of complicated data structures, and has been proved to be very efficient, from both points of view of theoretical complexity and pratical results. We take particular interest in designing dynamic algorithms : in practice, the data of a problem are often acquired progressively. It is obviously not reasonable to compute the whole result again each time a new data is inserted, (semi-) dynamic schemes are thus necessary. We introduce a very general data structure, the influence graph, that allows us to construct various geometric structures : Voronoi diagrams, arrangements of line segments... We study both theoretical complexity and pratical efficiency of the algorithms

From triangles to curves

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Computing a Dirichlet Domain for a Hyperbolic Surface

This paper exhibits and analyzes an algorithm that takes a given closed orientable hyperbolic surface and outputs an explicit Dirichlet domain. The input is a fundamental polygon with side pairings. While grounded in topological considerations, the algorithm makes key use of the geometry of the surface. We introduce data structures that reflect this interplay between geometry and topology and show that the algorithm runs in polynomial time, in terms of the initial perimeter and the genus of the surface

Triangulations in CGAL - To non-Euclidean spaces and beyond!

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