We 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.

中文翻译：

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非流形三角形网格的拉普拉斯算子

我们描述了适用于任何三角形网格的离散拉普拉斯算子，包括那些非流形或不可定向（有边界或无边界）的三角形网格。我们的 Laplacian 是通常 cotan 矩阵的强大替代品，并且保证在内部和边界边缘上具有非负边缘权重，即使对于质量极差的网格也是如此。关键思想是在输入域上构建我们所说的“簇状覆盖”，它具有非流形顶点但有流形边。由于所有边都是流形的，我们可以翻转到内在的 Delaunay 三角剖分；我们的拉普拉斯算子就是这个新三角剖分的 cotan 拉普拉斯算子。这种构造还通过点集的非流形三角剖分提供了高质量的点云拉普拉斯算子。我们在各种具有挑战性的示例（包括来自 Thingi10k 的所有模型）上验证我们的拉普拉斯算子，