Search for: All records

This paper introduces a method to synthesize a 3D tensor field within a constrained geometric domain represented as a tetrahedral mesh. Whereas previous techniques optimize forisotropicfields, we focus onanisotropictensor fields that are smooth and aligned with the domain boundary or user guidance. The key ingredient of our method is a novel computational design framework, built on top of thesymmetric orthogonally decomposable(odeco) tensor representation, to optimize the stretching ratios and orientations for each tensor in the domain. In contrast to past techniques designed only forisotropictensors, we demonstrate the efficacy of our approach in generating smooth volumetric tensor fields with highanisotropyand shape conformity, especially for the domain with complex shapes. We apply these anisotropic tensor fields to various applications, such as anisotropic meshing, structural mechanics, and fabrication. 

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 coarseintrinsic triangulationof the input domain. In the spirit of thequadric 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.