  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.
