Sharp, NicholasCrane, KeenanJacobson, Alec and Huang, Qixing2020-07-052020-07-0520201467-8659https://doi.org/10.1111/cgf.14069https://diglib.eg.org:443/handle/10.1111/cgf14069We describe a discrete Laplacian suitable for any triangle mesh, including those that are nonmanifold or nonorientable (with or without boundary). Our Laplacian is a robust drop-in replacement for the usual cotan matrix, and is guaranteed to have nonnegative edge weights on both interior and boundary edges, even for extremely poor-quality meshes. The key idea is to build what we call a ''tufted cover'' over the input domain, which has nonmanifold vertices but manifold edges. Since all edges are manifold, we can flip to an intrinsic Delaunay triangulation; our Laplacian is then the cotan Laplacian of this new triangulation. This construction also provides a high-quality point cloud Laplacian, via a nonmanifold triangulation of the point set. We validate our Laplacian on a variety of challenging examples (including all models from Thingi10k), and a variety of standard tasks including geodesic distance computation, surface deformation, parameterization, and computing minimal surfaces.Attribution 4.0 International LicenseMathematics of computingDiscretizationPartial differential equations10.1111/cgf.1406969-80