Constructing L∞ Voronoi Diagrams in 2D and 3D

Voronoi diagrams and their computation are well known in the Euclidean L-2 space. They are easy to sample and render in generalized L-p spaces but nontrivial to construct geometrically. Especially the limit of this norm with p -> infinity lends itself to many quad- and hex-meshing related applications as the level-set in this space is a hypercube. Many application scenarios circumvent the actual computation of L-infinity diagrams altogether as known concepts for these diagrams are limited to 2D, uniformly weighted and axis-aligned sites. Our novel algorithm allows for the construction of generalized L-infinity Voronoi diagrams. Although parts of the developed concept theoretically extend to higher dimensions it is herein presented and evaluated for the 2D and 3D case. It further supports individually oriented sites and allows for generating weighted diagrams with anisotropic weight vectors for individual sites. The algorithm is designed around individual sites, and initializes their cells with a simple meshed representation of a site's level-set. Hyperplanes between adjacent cells cut the initialization geometry into convex polyhedra. Non-cell geometry is filtered out based on the L-infinity Voronoi criterion, leaving only the non-convex cell geometry. Eventually we conclude with discussions on the algorithms complexity, numerical precision and analyze the applicability of our generalized L-infinity diagrams for the construction of Centroidal Voronoi Tessellations (CVT) using Lloyd's algorithm.

Automated summary

Voronoi diagrams and their computation are well known in the Euclidean L-2 space. Our novel algorithm allows for the construction of generalized L-infinity Voronoi diagrams, supporting individually oriented sites and generating weighted diagrams with anisotropic weight vectors for individual sites.