Geodesic Voronoi Diagrams with Polyline Generators

Geodesic Voronoi diagrams (GVDs) defined on triangle meshes with polyline generators are studied in this paper. We introduce a new concept, called local Voronoi diagram, or LVD, which is a weighted Euclidean Voronoi diagram on a mesh triangle. We show that when restricting on a mesh triangle, the GVD is a subset of the LVD, which can be computed by using the existing 2D techniques. Moreover, only two types of mesh faces can contain GVD edges. Guided by our theoretical findings, the geodesic Voronoi diagram with polyline generators can be built in O ( nN log N ) time and takes O ( nN ) space on an n-face mesh with m generators, where N = max { m , n } .