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Laplace-Beltrami Eigenfunctions Towards an Algorithm That "Understands" Geometry
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Discrete conformal mappings via circle patterns
ACM Transactions on Graphics (TOG)
Discrete laplace operators: no free lunch
SGP '07 Proceedings of the fifth Eurographics symposium on Geometry processing
A Monotonicity Property for Weighted Delaunay Triangulations
Discrete & Computational Geometry
Conformal equivalence of triangle meshes
ACM SIGGRAPH 2008 papers
Discrete laplace operator on meshed surfaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
IEEE Transactions on Visualization and Computer Graphics
Discrete Laplace--Beltrami operators and their convergence
Computer Aided Geometric Design
Laplace-Beltrami spectra as 'Shape-DNA' of surfaces and solids
Computer-Aided Design
Global intrinsic symmetries of shapes
SGP '08 Proceedings of the Symposium on Geometry Processing
A concise and provably informative multi-scale signature based on heat diffusion
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Convergence, stability, and discrete approximation of Laplace spectra
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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Computer Aided Geometric Design
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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.