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.

翻译：我们描述一种有效的算法,用以计算离散表面(可能包括边界)的相容等量度,在边界沿线的所有内部脊椎和指定的大地测量曲线上展示规定的高斯曲线。我们的构造基于[Gu等人,2018年;Springborn,2020年]所开发的理论,特别是依赖双曲三角测量结果。一般化是通过将表面固有的三角测量视为自由程度来实现的,并特别关注对地表边界的正确处理。虽然通过双层覆盖法,边界案可以自然地被简化为封闭案件,但设定的隐含对称对称会增加与我们在这项工作中明确处理的稳定的德拉纳关键配置有关的挑战。