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移动坐标系方法的表面全局参数化的新表示方法。我们证明结构方程组，表征相对于局部框架系统的局部坐标变化，完全表征了可能的圆锥参数化集。事实证明，该系统的离散化为有效映射的存在提供了充分必要条件。我们能够导出一种用于表面参数化的通用算法，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，因此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圆锥度量拓扑之间的联系来解决具有挑战性的约束参数化问题。