A new spatial data structure for triangular mesh with toroidal patches

Abstract

We introduce a novel spatial data structure for a triangular mesh consisting of osculating toroidal patches. A toroidal patch is generated using the differential geometry of a vertex by approximating a local surface to estimate the principal curvatures and directions. Within a given tolerance, the adjacent faces of a vertex can effectively be approximated by the toroidal patch. For efficient usage, toroidal patches are constructed in a hierarchical manner using various tolerance values ϵ=10−3,10−4, and 10−5 and combined with the conventional Bounding Volume Hierarchy (BVH). To demonstrate the effectiveness of our approach, we employ our spatial data structure for filling holes in triangular meshes. After generating a hole-filled patch, we update the hole-filled patch’s vertices using the normals and positions of the toroidal patches. To the best of our knowledge, this is the first application of toroidal patches to triangular mesh.