Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams

Abstract

We discuss the following problem: given n points in the plane (the "sites"), and an arbitrary query point q, find the site that is closest to q. This problem can be solved by constructing the Voronoi diagram of the given sites, and then locating the query point in one of its regions. We give two algorithms, one that constructs the Voronoi diagram in O(n lg n) time, and another that inserts a new site in O(n) time. Both are based on the use of the Voronoi dual, the Delaunay triangulation, and are simple enough to be of practical value. The simplicity of both algorithms can be attributed to the separation of the geometrical and topological aspects of the problem, and to the use of two simple but powerful primitives, a geometric predicate and an operator for manipulating the tgpology of the diagram. The topology is represented by a new data structure for generalized diagrams, that is embeddings of graphs in two-dimensional manifolds. This structure represents simultaneously an embedding, its dual, and its mirror-image. Furthermore, just two operators are sufficient for building and modifying arbitrary diagrams. O. IntroductionOne of the ft,ndamental data structures of computational geometry is the Voronoi diagram. This diagram arises from consideration of the following natural problem. Let n points in the plane be given, called sites. We wish to preprocess them into a data structure, so that given a new query point q, we can efficiently locate the nearest neighbor of q among the sites. The n sites in fact partition the plane into a collection of n regions, each associated with one of the sites. If region P is associated with site p, then P is the locus of all points in the plane closer to p than to any of the other n --1 sites.This partition is known as the Voronoi diagram (or the Dirichlet, or Thiessen, tesselation) determined by the given sites.The closest site problem is can therefore be solved by constructing the Voronoi diagram, and then locating the query point in it. Using the currently best available algorithms, the Voronoi diagram of n points can be computed in O(nlgn) time and stored in O(n) space; these bounds have been shown to be optimal in the worst case [Sh]. Once we have the Voronoi diagram, we can construct in linear further time a structure with which we can do point location in a planar subdivision in O(lg n) time [Kil].The work of Jorge Stolfi. who is on leave from the University of Sit Paulo (Sit Paulo, Brazil) was partially s'upponed by a grant from Conselho Nacional de Desenvolvimento Cienlffioo e Tecnol6gico (CNPq).Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission.© 1983 ACM 0-89791-099-0/83/004/0221 $00.75Shamos [Sh] first pointed out that ...

Morphological mesh filtering and -objects

“…Guibas and J. Stolfi (Guibas & Stolfi, 1985) have already proposed an algebraic structure to create and manipulate any kind of geometrical mesh. Quadedge data structure is a particularly elegant data structure for polyhedra.…”

“…3D reconstruction process of an obstacle (the chair) using the designed mesh analysis operators. From left to right : Original image with a zoom on a chair-shaped cluster/obstacle -Stereo-reconstruction of the cluster associated to the obstacle -Projection of the 3D point set onto its eigen plane and morphological mesh opening -Two views of the 3D retroprojection of the opened mesh (Guibas & Stolfi, 1985) where edges play a leading role. It allows an edge-to-edge navigation through the mesh by means of its algebraic operations.…”

Voronoi diagram: An adaptive spatial tessellation for processes simulation

“…Therefore, to find the topological event of a moving point, only the spatial information of the triangles/tetrahedra having the moving point as one of their vertexes and their neighbors are used and the remaining triangles/tetrahedra in the tessellation do not need to be tested. This can be computed using well-known predicted test (Guibas and Stolfi, 1985) to preserve the Delaunay empty circumcircle/circumsphere criterion. Since in a kinetic data structure, the position of points are time dependent, then, the value of the determinant will be time dependent as well.…”

Breps as Displayable-Selectable Models in Interactive Design of Families of Geometric Objects

“…A large number of research papers have been written about Breps. Most of these papers have dealt with the design and analysis of specific Brep data structures [Woo85,Mäntylä88,Alla91,Guibas85] and with algorithms operating on Breps, such as Boolean operations [Requicha85,Hoffmann89]. The essence of a Brep is best described as explicitly representing open, dimensionally uniform, connectivity components of intersection entities generated by a set of geometric pointset carriers.…”