The geometry of circles: Voronoi diagrams, Möbius transformations, Convex Hulls, Fortune’s algorithm, the cut locus and parametrization of shapes
نویسندگان
چکیده
The geometry of circles in the plane is inextricably tied with the group of Möbius transformations, which take circles to circles. This geometry can be seen in a more symmetric after transforming the plane to the sphere, by stereographic projection. Interpretations will be discussed for Voronoi diagrams, Delaunay triangulations,etc. from this point of view. Fortune’s algorithm for constructing the Voronoi diagram may be interpreted as a method of calculating the intersection of a set of halfspaces bounded by planes tangent to a sphere. The convex hull is ordered in terms of inclusion in the “horizon cone” for a point moving along a line tangent to the sphere. Voronoi diagrams and convex hulls are closely related to the cut locus for a curve (or more general set) in the plane. A modification of Fortune’s algorithm for calculating the cut locus of a smooth curve is discussed.
منابع مشابه
Metric Combinatorics of Convex Polyhedra: Cut Loci and Nonoverlapping Unfoldings
Let S be the boundary of a convex polytope of dimension d + 1, or more generally let S be a convex polyhedral pseudomanifold. We prove that S has a polyhedral nonoverlapping unfolding into R, so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in R by identifying pairs of boundary faces isometrically. Our existence proof exploits geodesic flow away from a source ...
متن کاملAn Optimal Algorithm for Intersecting Three-Dimensional Convex Polyhedra
This paper describes a linear-time algorithm for computing the intersection of two convex polyhedra in 3-space. Applications of this result to computing intersections, convex hulls, and Voronoi diagrams are also given.
متن کاملCalculating Voronoi Diagrams using Convex Sweep Curves
The Voronoi diagram is an important data structure in computational geometry. Given n sites in the plane, the Voronoi diagram partitions the plane into n regions. The region of a site p consists of all those points that lie closer to p than to any of the other sites. For a survey on Voronoi diagrams and their applications we refer to Aurenhammer [1]. A generalization of the sweep line method of...
متن کاملAn Efficient Implementation of Fortune’s Plane-Sweep Algorithm for Voronoi Diagrams
1 Introduction
متن کاملComputational geometry for curved objects: Voronoi diagrams in the plane
We examine the problem of computing exactly the Delaunay graph (and the dual Voronoi diagram) of a set of, possibly intersecting, smooth convex pseudo-circles in the Euclidean plane, given in parametric form. Pseudo-circles are (convex) closed curves, every pair of which has at most two intersection points. We propose robust end efficient algorithms for all required predicates under the exact c...
متن کامل