H.B. Keller Colloquium

Whether it be for geometry processing or finite element analysis, computations over surface meshes rely heavily on the discretization (weak form) of differential operators such as gradient, Laplacian, covariant derivative, or shape operator. While a variety of discrete operators over triangulated meshes have been developed and used for decades, a similar construction over polygonal meshes remains far less explored despite the prevalence of non-simplicial surfaces in geometric design and engineering applications. This paper introduces a principled construction of discrete differential operators on surface meshes formed by (possibly non-flat and non-convex) polygonal faces. Our approach is based on a novel mimetic discretization of the gradient operator that is linear-precise on arbitrary polygons. From this discrete gradient, we draw upon ideas from Discrete Exterior Calculus and the Virtual Element Method in order to derive a series of discrete operators commonly needed in graphics that are now valid over polygonal surfaces. We demonstrate the accuracy and robustness of our resulting operators by analyzing the convergence of residual errors on polygonal meshes with increasing resolution, and by applying the resulting operators on archetypical geometry processing tasks including mesh parameterization, fast geodesic distance evaluation, grooming (vector field editing), and shape editing. (Joint work with Dr. Fernando de Goes and Andrew Butts, Pixar Animation Studios.)