Given a triangulated piecewise-flat surface and a function on the vertices we can define the Dirichlet energy, which is related to the Dirichlet energy of a smooth function. To find the Delaunay triangulation of the piecewise flat surface, we modify the triangulation by a sequence of edge flips, called the flip algorithm, which transform an edge which is not Delaunay into one that is Delaunay. It is known that the flip algorithm works in the plane as well as for a piecewise-flat surface, where we have to ensure that only finitely many triangulations are possible.

When the vertices of a piecewise-flat surface have weights, we want to find the weighted Delaunay triangulation using a flip algorithm. In this dissertation, we prove that the maximum edge length during the algorithm is bounded, which guarantees that there are finitely many triangulations. Thus the flip algorithm terminates and the resulting triangulation is weighted Delaunay.

Additionally, we give a new way to find what we call the relaxed weighted Delaunay on a flat surface.