A Monotonicity Property for Weighted Delaunay Triangulations

where i is the linear interpolation of f over the triangle Ti in T and the sum is over all triangles in the triangulation. One may consider changing the triangulation by exchanging two triangles joined by an edge, forming a quadrilateral, by the triangles obtained by switching the diagonal of the quadrilateral; this is called an edge ‡ip or a 2 ! 2 bistellar ‡ip. He showed that the roughness of a triangulation decreases when an edge is ‡ipped to make the edge Delaunay. Since every triangulation can be transformed into a Delaunay triangulation by a sequence of edge ‡ips, this implies that the roughness is minimized by a Delaunay triangulation. The roughness is equal to the following functional, which we call the Dirichlet energy: E (f; T ) = 1