Computing Harmonic Maps and Conformal Maps on Point Clouds (Journal of Computational Mathematics, 2023)
We use a narrow-band approach to compute harmonic maps and conformal maps forsurfaces 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 $ε$-neighborhood. Then the harmonic map of the surface can be approximated by discreteharmonic maps on lattices. The conformal map, or the surface uniformization, is achievedby minimizing the Dirichlet energy of the harmonic map while deforming the target surfaceof constant curvature. We propose algorithms and numerical examples for closed surfacesand topological disks. To the best of the authors’ knowledge, our approach provides thefirst meshless method for computing harmonic maps and uniformizations of higher genussurfaces.