离散 Laplace 算子——球面与双曲情形
Discrete Laplacians -- spherical and hyperbolic
AI总结：
本文定义球面与双曲三角曲面上的离散 Laplace 算子，保持 Delaunay 非负权、面积表示、共形特征函数及无穷小等距变形等几何结构。
摘要：
欧氏三角剖分曲面上的离散 Laplace 算子已是一个成熟概念。我们引入球面与双曲三角剖分曲面上的离散 Laplace 算子。一方面，我们的定义与欧氏情形相近：边权包含某些角度组合的余切，并且边权非负当且仅当该三角剖分是 Delaunay 三角剖分。另一方面，这些离散化在若干方面保持结构。我们证明，凸多面体的面积可以用支撑函数的离散球面 Laplace 算子表示，其表达式与光滑凸体的面积借助通常球面 Laplace 算子的表达式相同。我们表明，在曲率为 $k \in \{-1,1\}$ 的三角剖分曲面上，离散共形向量场的共形因子是我们所给离散 Laplace 算子的 $-2k$ 特征函数，这与光滑情形完全一致。这里的离散共形性既可以按顶点缩放理解，也可以按圆模式理解。最后，我们将这些 $-2k$ 特征函数与内接于相应二次曲面的多面体的无穷小等距变形联系起来。
英文摘要：
The discrete Laplacian on Euclidean triangulated surfaces is a well-established notion. We introduce discrete Laplacians on spherical and hyperbolic triangulated surfaces. On the one hand, our definitions are close to the Euclidean one in that the edge weights contain the cotangents of certain combinations of angles and are non-negative if and only if the triangulation is Delaunay. On the other hand, these discretizations are structure-preserving in several respects. We prove that the area of a convex polyhedron can be written in terms of the discrete spherical Laplacian of the support function, whose expression is the same as the area of a smooth convex body in terms of the usual spherical Laplacian. We show that the conformal factors of discrete conformal vector fields on a triangulated surface of curvature $k \in \{-1,1\}$ are $-2k$-eigenfunctions of our discrete Laplacians, exactly as in the smooth setting. The discrete conformality can be understood here both in the sense of the vertex scaling and in the sense of circle patterns. Finally, we connect the $-2k$-eigenfunctions to infinitesimal isometric deformations of a polyhedron inscribed into corresponding quadrics.
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