This paper describes a method for fast simplification of surface meshes. Whereas past methods focus on visual appearance, our goal is to solve equations on the surface. Hence, rather than approximate the extrinsic geometry, we construct a coarse intrinsic triangulation of the input domain. In the spirit of the quadric error metric (QEM), we perform greedy decimation while agglomerating global information about approximation error. In lieu of extrinsic quadrics, however, we store intrinsic tangent vectors that track how far curvature "drifts" during simplification. This process also yields a bijective map between the fine and coarse mesh, and prolongation operators for both scalar- and vector-valued data. Moreover, we obtain hard guarantees on element quality via intrinsic retriangulation - a feature unique to the intrinsic setting. The overall payoff is a "black box" approach to geometry processing, which decouples mesh resolution from the size of matrices used to solve equations. We show how our method benefits several fundamental tasks, including geometric multigrid, all-pairs geodesic distance, mean curvature flow, geodesic Voronoi diagrams, and the discrete exponential map.

翻译：本文描述了一种快速简化曲面网格的方法。与以往方法聚焦于视觉表现不同，我们的目标是求解曲面上的方程。因此，我们并非逼近外在几何，而是构建输入域的一个粗糙内在三角剖分。遵循二次误差度量（QEM）的思想，我们在贪婪精简过程中聚合关于近似误差的全局信息。然而，我们并非使用外在二次型，而是存储内在切向量，以追踪简化过程中曲率的“漂移”。这一过程还生成了精细网格与粗糙网格之间的双射映射，以及适用于标量和向量值数据的延拓算子。此外，通过内在重三角剖分（内在设定下独有的特征），我们获得了对单元质量的硬性保证。总体成效是一种“黑盒”几何处理方法，它将网格分辨率与求解方程所用矩阵的规模解耦。我们展示了该方法如何惠及若干基础任务，包括几何多重网格、全对测地距离、平均曲率流、测地Voronoi图以及离散指数映射。