A Hands-On Introduction to Discrete Differential Operators on Polygon Meshes

Many applications in geometry processing involve the solution of partial differential equations on discrete surface meshes, with the Laplacian undoubtedly being the most ubiquitous operator in this context. Having discrete operators for gradient, divergence, and Laplacian at hand allows to solve many interesting geometry processing problems. Unfortunately, many approaches or implementations require the mesh to be a well-behaved triangle mesh with good-quality elements, and severely degrade or completely fail if these conditions are not met. In this tutorial, we will present how to discretize (and implement) gradient, divergence, and Laplacian operators in a simple, flexible, and robust manner. The presented discrete differential operators can be applied to triangle meshes, quad meshes, or general polygon meshes, they work robustly even for low-quality or degenerate elements, and as such, they allow to generalize many geometry processing algorithms to a much wider range of mesh inputs. We also provide interactive {HTML}-based course notes at https://graphics.rocks/eg26DDG .