The complexity of bisectors and Voronoi diagrams on realistic terrains

We prove tight bounds on the complexity of bisectors and Voronoi diagrams on so-called realistic terrains, under the geodesic distance. In particular, if n denotes the number of triangles in the terrain, we show the following two results. (i) If the triangles of the terrain have bounded slope and the projection of the set of triangles onto the xy-plane has low density, then the worst-case complexity of a bisector is T(n). (ii) If, in addition, the triangles have similar sizes and the domain of the terrain is a rectangle of bounded aspect ratio, then the worst-case complexity of the Voronoi diagram of m point sites is T(n+mvn