A Short Course in Computational Geometry and Topology by Herbert Edelsbrunner

Author:Herbert Edelsbrunner
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

(7.3)

Fig. 7.4The Voronoi diagram of sites and the full subcomplex of the alpha complex defined by the four sites whose disks contain the point

Since this is true for every point in the plane, we can set and compute the area of the union of disks by integration, as before. To collect the contributions, we iterate over all vertices, edges, and triangles of :

Short PIE Formula The area of the union of disks of radius centered at the sites in is

In the plane, has at most a constant times simplices, and each simplex corresponds to an independent collection of disks. We thus have a formula with few and simple terms. The formula generalizes to higher dimensions, and to collections of balls with different radii [3]. The summation can also be done over all simplices of the Delaunay triangulation, and while this sum contains redundant terms, the result is still correct [4].

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.