🤖 AI Summary
This paper addresses topological ambiguities and numerical robustness issues in Constructive Solid Geometry (CSG) Boolean operations and mesh repair—arising from non-manifold intersections, multi-operand expressions, and degenerate geometries (e.g., coplanar or collinear features). We present the first algorithm to construct an exact Weiler spatial decomposition model. Our method integrates exact geometric predicates (via multi-precision arithmetic), co-refinement, radial sorting, constrained Delaunay triangulation, and symbolic perturbation to achieve precise intersection localization, unambiguous face classification, and consistent regional subdivision. Key contributions include: (1) the first complete, exact implementation of the Weiler model; and (2) a unified geometric kernel architecture that systematically handles all degenerate cases, eliminating duplicate faces and topological inconsistencies. Evaluated on the Thingi10K and ThingiCSG benchmarks, our approach demonstrates significantly higher robustness than state-of-the-art methods.
📝 Abstract
This article introduces a general mesh intersection algorithm that exactly computes the so-called Weiler model 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. This is, to our knowledge, the first algorithm that computes an exact Weiler model. To implement all the involved predicates and constructions, two geometric kernels 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"