Direct Rendering of Intrinsic Triangulations

Abstract

Existing intrinsic triangulation frameworks represent powerful tools for geometry processing; however, they all require the extraction of the common subdivision between extrinsic and intrinsic triangulations for visualization and optimized data transfer. We describe an efficient and effective algorithm for directly rendering intrinsic triangulations that avoids extracting common subdivisions. Our strategy is to use GPU shaders to render the intrinsic triangulation while rasterizing extrinsic triangles. We rely on a point-location algorithm supported by a compact data structure, which requires only two values per extrinsic triangle to represent the correspondence between extrinsic and intrinsic triangulations. This data structure is easier to maintain than previous proposals while supporting all the standard topological operations for improving the intrinsic mesh quality, such as edge flips, triangle refinements, and vertex displacements. Computational experiments show that the proposed data structure is numerically robust and can process nearly degenerate triangulations. We also propose a meshless strategy to accurately transfer data from intrinsic to extrinsic triangulations without relying on the extraction of common subdivisions.