Abstract: The Voronoi diagram is a decomposed region X made into cells, the decomposition being induced by a set of generators (Pk)k-K, and a distance function, and involves finding for each generator Pk a cell, which is a set of all the points in X satisfying the condition that the distance to the current generator P=Pk is not greater than the distance thereof to the union A of the other generators. For each generator, and for each point p in this generator, the method selects a set of directions, then for each direction recursively tests a ray in that direction, until a certain interval on the ray is shorter than or equal to a given error parameter. A point corresponding to the interval on the ray is then selected as an end point, and cells are defined from the end points, thus forming the Voronoi diagram.

Abstract: A method of producing and storing a Voronoi diagram includes: a) selecting a desired site Pk and a desired point p in the site; b) selecting a plurality of subfaces corresponding to a manifold around said point p; c) for each desired subface selecting a plurality of directions; d) for each direction providing a ray, selecting a point on the ray as an endpoint, and selecting hyperplanes corresponding to the endpoints; e) for each desired subface determining a type of intersection between the detected hyperplanes; f) determining whether there exist vertices of the cell in a cone generated by the endpoints, and finding said vertices, wherein if there remain further vertices that have not been found then there is carried out a step of further dividing the said subface into additional subfaces for separate determination; g) storing at least one of the following: said vertices, hyperplanes, neighbor sites, and possibly other information such as endpoints; h) repeating a)-g) for each desired site Pk and each desired

Abstract: A computerized method of computing the Voronoi diagram has applications including communications networks, robotics, three-dimensional networks, materials science, searching image processing, data clustering, data compression, control of a groups of methods for image processing and the like, design of electronic circuits, geographic information systems, solutions of the efficient location problem, face recognition, mesh generation and re-meshing, curve and surface generation/reconstruction, solid modeling, collision detection, controlling motion of vehicles, navigation, accident prevention, data clustering and data processing, proximity operations, nearest neighbor search, numerical simulations, weather prediction, analyzing and modeling proteins and other biological structures, designing drugs, finding shortest paths, pattern recognition and as an artistic tool.