Triangle Surfaces with Discrete Equivalence Classes

We propose a technique that takes a triangulated surface as input and outputs a surface with the same topology but altered geometry such that each polygon falls into a set of discrete equivalence classes. We begin by describing an error function that measures how close the polygons are to satisfying this criteria. To optimize this error function, we first cluster triangles into discrete sets such that the assignment of sets minimizes our error. We then find canonical polygons for each set using nonlinear optimization. Next, we solve a Poisson equation to find positions of vertices such that the surface polygons match the canonical polygons as close as possible. We also describe how to incorporate a fairness criteria into the optimization to avoid oscillations of the surface. We iterate this entire process until we reach a user specified tolerance, possibly adding clusters during iteration to guarantee convergence. We have been able to successfully reduce the number of unique triangles to lie within a small percentage of the total number of triangles in the surface and demonstrate our technique on various examples.