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Journals and Conferences
This paper presents a general framework for the design and randomized analysis of geometric algorithms. These algorithms are on-line and the framework provides general bounds for their expected space and time complexities when averaging over all permutations of the input data. The method is general and can be applied to various geometric problems. The power… (More)
This paper presents the main algorithmic and design choices that have been made to implement triangulations in the computational geometry algorithms library Cgal.
The Delaunay Tree is a hierarchical data structure that has been introduced in  and analyzed in [6, 4]. For a given set of sites S in the plane and an order of insertion for these sites, the Delaunay Tree stores all the successive Delaunay triangulations. As proved before, the Delaunay Tree allows the insertion of a site in logarithmic expected time and… (More)
The purpose of this paper is to present a new method to design exact geometric predicates in algorithms dealing with curved objects such as circular arcs. We focus on the comparison of the abscissae of two intersection points of circle arcs, which is known to be a difficult predicate involved in the computation of arrangements of circle arcs. We present an… (More)
This article introduces the absolute quadratic complex formed by all lines that intersect the absolute conic. If /spl omega/ denotes the 3 /spl times/ 3 symmetric matrix representing the image of that conic under the action of a camera with projection matrix P, it is shown that /spl omega/ /spl ap/ P/sup ~//spl Omega//sub /spl I.bar//P/sup ~T/ where V is… (More)
Given a triangulation in the plane or a tetrahedralization in 3-space, we investigate the efficiency of locating a point by walking in the structure with different strategies.
Boissonnat, J.-D. and M. Teillaud, On the randomized construction of the Delaunay tree, Theoretical Computer Science 112 (1993) 339-354. The Delaunay tree is a hierarchical data structure which is defined from the Delaunay triangulation and, roughly speaking, represents a triangulation as a hierarchy of balls. It allows a semidynamic construction of the… (More)
Though Delaunay triangulations are very well known geometric data structures, the problem of the robust removal of a vertex in a three-dimensional Delaunay triangulation is still a problem in practice.We propose a simple method that allows to remove any vertex even when the points are in very degenerate configurations. The solution is available in CGAL.
We present, in this paper, a new hierarchical data structure called the Delaunay tree. It is defined from the Delaunay triangulation and, roughly speaking, represents a triangulation as a hierarchy of balls. The Delaunay tree provides efficient solutions to several problems such as building the Delaunay triangulation of a finite set of n points in any… (More)
Thek-Delaunay tree extends the Delaunay tree introduced in , and . It is a hierarchical data structure that allows the semidynamic construction of the higher-order Voronoi diagrams of a finite set ofn points in any dimension. In this paper we prove that a randomized construction of thek-Delaunay tree, and thus of all the order≤k Voronoi diagrams, can… (More)