Crystal Voronoi diagram and its applications

This paper studies the multiplicatively weighted crystal-growth Voronoi diagram, which describes the partition of the plane into crystals with different growth speeds. This type of the Voronoi diagram is defined, and its basic properties are investigated. The analytic equation describing the boundary curve is given for a simple case. For the general case, an approximation algorithm is proposed. This algorithm is based on a finite-difference method, called a fast marching method, for solving a special type of a partial differential equation. The proposed algorithm is applied to the planning of a collision-free path for a robot avoiding enemy attacks.