Fast Discovery of Voronoi Vertices in the Construction of Voronoi Diagram of 3D Balls

Solving geometrical problems on a set of 3D balls is a challenging task in computational geometry. They can be solved effectively when the Voronoi diagram for the set is available. The diagram is usually constructed by the edge-tracing or similar algorithms based on finding Voronoi vertices along edges. However, its expected quadratic time complexity makes it impractical. This can be improved significantly by our new approach. Whenever a vertex needs to be found, Delaunay triangulation of ball centers is searched through to find one specific ball. The search is kept inside a spatial filter, which can be reduced in size during the search. The improvement is demonstrated on protein data (a set of balls represents atoms in a molecule), because this is our intended application.