Exact predicates, exact constructions and combinatorics for mesh CSG.

Abstract

This article introduces a general mesh intersection algorithm that exactly computes the so-called Weiler model (also called a 3D arrangement) and that uses it to implement boolean operations with arbitrary multi-operand expressions, CSG (constructive solid geometry) and some mesh repair operations. From an input polygon soup, the algorithm first computes the co-refinement, with an exact representation of the intersection points. Then, the decomposition of 3D space into volumetric regions (Weiler model) is constructed, by sorting the facets around the non-manifold intersection edges (radial sort), using specialized exact predicates. Finally, based on the input boolean expression, the triangular facets that belong to the boundary of the result are classified. The main contribution is a 2D Constrained Delaunay Triangulation with exact coordinates that represent the intersections, thanks to two geometric kernels that are proposed, tested and discussed (arithmetic expansions and multi-precision floating-point). As a guiding principle, the combinatorial information shared between each step is kept as simple as possible. It is made possible by treating all the particular cases in the kernel. In particular, triangles with intersections are remeshed using the (uniquely defined) Constrained Delaunay Triangulation, with symbolic perturbations to disambiguate configurations with co-cyclic points. It makes it easy to discard the duplicated triangles that appear when remeshing overlapping facets. The method is tested and compared with previous work, on the existing “thingi10K” dataset (to test co-refinement and mesh repair) and on a new “thingiCSG” dataset made publicly available (to test the full CSG pipeline) on a variety of interesting examples featuring different types of “pathologies”.