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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., Comment: SIGGRAPH 2023

In this work, we present a general, efficient, and provably robust representation for intrinsic triangulations. These triangulations have emerged as a powerful tool for robust geometry processing of surface meshes, taking a low-quality mesh and retriangulating it with high-quality intrinsic triangles. However, existing representations either support only edge flips, or do not offer a robust procedure to recover the common subdivision, that is, how the intrinsic triangulation sits along the original surface. To build a general-purpose robust structure, we extend the framework of normal coordinates, which have been deeply studied in topology, as well as the more recent idea of roundabouts from geometry processing, to support a variety of mesh processing operations like vertex insertions, edge splits, etc. The basic idea is to store an integer per mesh edge counting the number of times a curve crosses that edge. We show that this paradigm offers a highly effective representation for intrinsic triangulations with strong robustness guarantees. The resulting data structure is general and efficient, while offering a guarantee of always encoding a valid subdivision. Among other things, this allows us to generate a high-quality intrinsic Delaunay refinement of all manifold meshes in the challenging Thingi10k dataset for the first time. This enables a broad class of existing surface geometry algorithms to be applied out-of-the-box to low-quality triangulations.