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.

中文翻译：

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具有边界的有效且鲁棒的离散共形等价

我们描述了一种有效的算法，可为可能具有边界的离散曲面计算保形等效度量，该曲面在所有内部顶点均显示规定的高斯曲率，并在边界处显示规定的测地曲率。我们的构建基于[Gu et al。2018; [Springborn 2020]，尤其是依赖于双曲线Delaunay三角剖分的结果。通用性是通过将表面的固有三角剖分视为自由度来实现的，并特别注意对表面边界的正确处理。尽管可以通过双重覆盖方法将边界情况自然地减少为封闭情况，但设置的隐含对称性会带来与稳定Delaunay关键配置有关的其他挑战，我们将在本文中明确解决这一问题。