OPTIMAL VORONOI DIAGRAM CONSTRUCTION WITH n CONVEX SITES IN THREE DIMENSIONS

Abstract

This paper presents a worst-case optimal algorithm for constructing the Voronoi diagram for n disjoint convex and rounded semi-algebraic sites in 3 dimensions.Rather than extending optimal 2-dimensional methods, 32 ' 16 ' 20 ' 2 we base our method on a suboptimal 2-dimensional algorithm, outlined by Lee and Drysdale and modified by Sharir 20 ' 30 for computing the diagram of circular sites.For complexity considerations, we assume the sites have bounded complexity, i.e., the algebraic degree is bounded as is the number of algebraic patches making up the site. For the sake of simplicity we assume that the sites are what we call rounded. This assumption simplifies the analysis, though it is probably unnecessary.Our algorithm runs in time 0(C(n)) where C(n) is the worst-case complexity of an n-site diagram. For spherical sites C(n) is 9(n 2 ), but sharp estimates do not seem to be available for other classes of site.

Compact families of Jordan curves and convex hulls in three dimensions

“…We make the following initial assumptions, which have been invoked in a previous work [7]. Further conditions will be stated in the next section.…”

“…The features of H(S) are its facets, edges, and vertices, as follows. As discussed in [14,7], ∂H(S) is naturally divided into connected regions: its exposed facets, tunnel facets, and planar facets. The exposed facets are (path-) connected components of (∂H(S)) ∩ S, tunnel facets are connected part-surfaces generated by line-segments touching two bodies, and (since the bodies are in general position) planar facets are triangular.…”

“…Without loss of generality v 0 = (1, 0, 0), and v 1 = (0, 1, 0). 7 8.3 The pre-seam is a C 1 manifold…”

The predicates for the Voronoi diagram of ellipses

“…In [19], an optimal combinatorial algorithm is derived for constructing Voronoi diagrams of strictly convex rounded sites in R 3 , but the predicates are not considered. In [7], a dynamic algorithm is described for constructing the power diagrams of points in R d .…”