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Volume 41, Issue 5
Computing Harmonic Maps and Conformal Maps on Point Clouds

Tianqi Wu & Shing-Tung Yau

DOI: 10.4208/jcm.2206-m2020-0251

J. Comp. Math., 41 (2023), pp. 879-908.

Published online: 2023-05

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  • 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.

  • Keywords

harmonic maps, conformal maps, point clouds.

  • AMS Subject Headings

68U05

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

mike890505@gmail.com (Tianqi Wu)

yau@math.harvard.edu (Shing-Tung Yau)

  • BibTex
  • RIS
  • TXT
@Article{JCM-41-879, author = {Wu , Tianqi and Yau , Shing-Tung}, title = {Computing Harmonic Maps and Conformal Maps on Point Clouds}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {41}, number = {5}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2206-m2020-0251}, url = {http://global-sci.org/intro/article_detail/jcm/21678.html} }
TY - JOUR T1 - Computing Harmonic Maps and Conformal Maps on Point Clouds AU - Wu , Tianqi AU - Yau , Shing-Tung JO - Journal of Computational Mathematics VL - 5 SP - 879 EP - 908 PY - 2023 DA - 2023/05 SN - 41 DO - http://doi.org/10.4208/jcm.2206-m2020-0251 UR - https://global-sci.org/intro/article_detail/jcm/21678.html KW - harmonic maps, conformal maps, point clouds. AB -

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.

Tianqi Wu & Shing-Tung Yau. (2023). Computing Harmonic Maps and Conformal Maps on Point Clouds. Journal of Computational Mathematics. 41 (5). 879-908. doi:10.4208/jcm.2206-m2020-0251
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