A Review of Properties and Variations of Voronoi Diagrams

32 ADAM DOBRIN7. The Centroidal Voronoi Diagram7.1. Introduction. A centroidal Voronoi diagram, or tessellation, isa Voronoi diagram of a given set such that every generator point isalso the centroid, or center of mass, of its Voronoi region. Typically,centroidal Voronoi tessellations (CVT’s) have an associated densityfunction, ρ(x, y), on Ω, a subset of R 2 .7.2. Real-World Applications and Observations. Applications forCVT’s cover a multitude of disciplines from computer science to urbanplanning.An interesting area in which CVT’s pop up is in the territorial behaviorof animals. For example, the male mouthbreeder fish, to establishdomain, will spit sand away from the center of its territory. For a highenough density of fish introduced simultaneously into a body of waterwith a uniform sandy floor, we find an interesting result. The spittingof sand results in raised bars of sand, and, when viewed from above,creates a pattern which very closely approximates a centroidal Voronoitessellation [3].Also in biology, CVT’s appear in the cell division of animal andplant cells. In a study of the development of starfish embryos, it wasfound that a view of a layer of columnar cells in an arrangement in a