## Abstract

The Laplace-Beltrami operator of a smooth Riemannian manifold is determined by the Riemannian metric. Conversely, the heat kernel constructed from the eigenvalues and eigenfunctions of the Laplace-Beltrami operator determines the Riemannian metric. This work proves the analogy on Euclidean polyhedral surfaces (triangle meshes), that the discrete heat kernel and the discrete Riemannian metric (unique up to a scaling) are mutually determined by each other. Given a 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 triangle mesh from a discrete Laplace-Beltrami operator matrix.

Original language | English (US) |
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Pages (from-to) | 121-129 |

Number of pages | 9 |

Journal | Graphical Models |

Volume | 74 |

Issue number | 4 |

DOIs | |

State | Published - Jul 2012 |

## All Science Journal Classification (ASJC) codes

- Software
- Modeling and Simulation
- Geometry and Topology
- Computer Graphics and Computer-Aided Design

## Keywords

- Discrete Riemannian metric
- Discrete curvature flow
- Discrete heat kernel
- Laplace-Beltrami operator
- Legendre duality principle