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 analogy on Euclidean polyhedral surfaces (triangle meshes), 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. First, we show that the space of all possible metrics of
a polyhedral surface is convex. Then, we construct a special energy defined on
the metric space, such that the gradient of the energy equals to the edge
weights. Third, we show the Hessian matrix of the energy is positive definite,
restricted on the tangent space of the metric space, therefore the energy is
convex. Finally, by the fact that the parameter on a convex domain and the
gradient of a convex function defined on the domain have one-to-one
correspondence, we show the edge weights determines the polyhedral metric
unique up to a scaling. The constructive proof leads to a computational
algorithm that finds the unique metric on a topological triangle mesh from a
discrete Laplace-Beltrami operator matrix.

本文讨论了 Laplace-Beltrami 算子在离散的欧几里得多面体表面的应用，证明了离散的 Laplace-Beltrami 算子和离散的 Riemannian 度量可以互相确定，并通过构建特殊能量和变分方法提出了求解算法。