Many parametrization and mapping-related problems in geometry processing can be viewed as metric optimization problems, i.e., computing a metric minimizing a functional and satisfying a set of constraints, such as flatness. Penner coordinates are global coordinates on the space of metrics on meshes with a fixed vertex set and topology, but varying connectivity, making it homeomorphic to the Euclidean space of dimension equal to the number of edges in the mesh, without any additional constraints imposed, and reducing to logarithms of edge lengths when restricted to a fixed connectivity. These coordinates play an important role in the theory of discrete conformal maps, enabling recent development of highly robust algorithms with convergence and solution existence guarantees for computing such maps. We demonstrate how Penner coordinates can be used to solve a general class of problems involving metrics, including optimization and interpolation, while retaining the key guarantees available for conformal maps.
翻译:在几何处理中,许多对称和绘图问题可视为衡量优化问题,即计算一个指标,最大限度地减少功能性和满足一系列制约因素,如平坦度。平纳座标是具有固定的脊椎板和地形学但互连互通程度各异的关于螺旋藻测量空间的全球坐标,使其与欧几里得空间的内形空间等同,不施加任何额外限制,在限制固定连通性时减少边缘长度的对数。这些座标在离散一致地图理论中发挥重要作用,使最近开发的高度稳健的算法能够在计算这些地图时有趋同和解决办法的保证。我们演示如何利用平纳座坐标解决涉及计量的一般问题,包括优化和内插,同时保留对符合要求的地图提供的关键保证。