Efficient and Robust Discrete Conformal Equivalence with Boundary

Abstract: We describe an efficient algorithm to compute a conformally equivalent metric for a discrete surface, possibly with boundary, exhibiting prescribed Gaussian curvature at all interior vertices and prescribed geodesic curvature along the boundary. Our construction is based on the theory developed in [Gu et al. 2018; Springborn 2020], and in particular relies on results on hyperbolic Delaunay triangulations. Generality is achieved by considering the surface's intrinsic triangulation as a degree of freedom, and particular attention is paid to the proper treatment of surface boundaries. While via a double cover approach the boundary case can be reduced to the closed case quite naturally, the implied symmetry of the setting causes additional challenges related to stable Delaunay-critical configurations that we address explicitly in this work.