Discrete Laplace-Beltrami Operator Determines Discrete Riemannian Metric

The Laplace-Beltrami operator of a smooth Riemannian manifold is determined by the Riemannian metric. Conversely, the heat kernel constructed from its eigenvalues and eigenfunctions determines the Riemannian metric. This work proves the discrete analogy on Euclidean polyhedral surfaces that the discrete Laplace-Beltrami operator and the discrete Riemannian metric (unique up to a scaling) are mutually determined by each other. Given an Euclidean polyhedral surface, its Riemannian metric is represented as edge lengths, satisfying triangle inequalities on all faces. The Laplace-Beltrami operator is formulated using the cotangent formula, where the edge weight is defined as the sum of the cotangent of angles against the edge. We prove that the edge lengths can be determined by the edge weights unique up to a scaling using the variational approach. The constructive proof leads to a computational algorithm that finds the unique metric on a topological triangle mesh from a discrete Laplace...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Xianfeng David Gu, Ren Guo, Feng Luo 0002, Wei Zeng
• Keywords: CORR 2010, Education, Euclidean Polyhedral, Laplace-Beltrami Operator, Riemannian Metric