Output Sensitive Construction of the Delaunay Triangulation of Points Lying in Two Planes

Abstract

In this paper, we propose an algorithm to compute the Delaunay triangulation of a set [Formula: see text] of n points in 3-dimensional space when the points lie in 2 planes. The algorithm is output-sensitive and optimal with respect to the input and the output sizes. Its time complexity is O(n log n+t), where t is the size of the output, and the extra storage is O(n).

Hyperbolic delaunay complexes and voronoi diagrams made practical

“…1 He gives two examples: the computation of 3D Delaunay complexes of sets lying in two planes [8], and optimal Möbius transformation and conformal mesh generation [5]. Hyperbolic geometry is also used in applications like graph drawing [29,24].…”

“…Several years ago, we showed that the hyperbolic Delaunay complex and Voronoi diagram can easily be deduced from their Euclidean counterparts [18,8]. As far as we know, this was the first time when the computation of hyperbolic Delaunay complexes and Voronoi diagrams was addressed.…”

“…Nielsen and Nock transform the computation of the Voronoi diagram in the non-conformal Klein model to the computation of a Euclidean power diagram [30]. Many other references can be found in [34] (which does not mention [18,8], though).…”

“…Due to this property, the model is used in a wide range of applications (see for instance [28,41,27]). We elaborate on our preliminary work [18,8], giving a detailed description of algorithms allowing to compute the hyperbolic Delaunay complex and Voronoi diagram in any dimension. All degenerate cases are handled.…”

Modelado, detección de colisiones y planificación de movimientos en sistemas robotizados mediante volúmenes esféricos

“…Esta representación de las esferas constrasta con otras representaciones utilizadas recientemente por otros autores como [Ped88], [DMT92], [BC+91], [BC+92] y [D&G94], que han representado una esfera como un punto en  4 (o un círculo como un punto en  3 ) mediante la 4-tupla (x,y,z,R) donde el radio de la esfera es r=x 2 +y 2 +z 2 -R. Si bien este tipo de representación tiene sus ventajas en la geometría computacional, no es así para el modelado que se presenta en este capítulo.…”