This article introduces a new representation of surface global parametrization based on Cartan’s method of moving frames. We show that a system of structure equations, characterizing the local coordinates changes with respect to a local frame system, completely characterizes the set of possible cone parametrizations. The discretization of this system provably provides necessary and sufficient conditions for the existence of a valid mapping. We are able to derive a versatile algorithm for surface parametrization, allowing feature constraints and singularities. As the first structure equation is independent of the global coordinate system, we do not require prior knowledge of cuts or cone positions. So, a single non-linear least-square problem is enough to place quantized cones while minimizing a given distortion energy. We are therefore able to take full advantage of the link between the parametrization geometry and the topology of its cone metric to solve challenging constrained parametrization problems.

本文介绍了一种基于嘉当动标架方法的曲面全局参数化的新表示。我们证明了一个结构方程系统，表征局部坐标相对于局部框架系统的变化，完全表征可能的锥体参数化集。该系统的离散化可证明为有效映射的存在提供了充分必要条件。我们能够推导出一种通用的表面参数化算法，允许特征约束和奇点。由于第一个结构方程独立于全局坐标系，我们不需要切割或锥体位置的先验知识。因此，单个非线性最小二乘问题足以放置量化的锥体，同时最小化给定的失真能量。因此，我们能够充分利用参数化几何与其锥度量的拓扑之间的联系来解决具有挑战性的约束参数化问题。