Year: 2023
Author: Tianqi Wu, Shing-Tung Yau
Journal of Computational Mathematics, Vol. 41 (2023), Iss. 5 : pp. 879–908
Abstract
We use a narrow-band approach to compute harmonic maps and conformal maps for surfaces embedded in the Euclidean 3-space, using point cloud data only. Given a surface, or a point cloud approximation, we simply use the standard cubic lattice to approximate its $\epsilon$-neighborhood. Then the harmonic map of the surface can be approximated by discrete harmonic maps on lattices. The conformal map, or the surface uniformization, is achieved by minimizing the Dirichlet energy of the harmonic map while deforming the target surface of constant curvature. We propose algorithms and numerical examples for closed surfaces and topological disks. To the best of the authors’ knowledge, our approach provides the first meshless method for computing harmonic maps and uniformizations of higher genus surfaces.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.2206-m2020-0251
Journal of Computational Mathematics, Vol. 41 (2023), Iss. 5 : pp. 879–908
Published online: 2023-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 30
Keywords: harmonic maps conformal maps point clouds.