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 - <p style="text-align: justify;">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.</p>