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 an 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. 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"

• The paper introduces an exact mesh intersection algorithm that accurately computes the Weiler model to partition 3D space.
• It employs Constrained Delaunay Triangulation and dual geometric kernels to handle arithmetic complexities and ensure robust intersection operations.
• Evaluated on extensive datasets, the approach shows superior reliability and performance for CAD, simulation, and visualization applications.

Overview of Robust Mesh CSG Operations

This paper presents a robust algorithm for performing mesh constructive solid geometry (CSG) operations with a significant emphasis on exact geometric computation. The focus is on designing an exact mesh intersection technique that computes the Weiler model—a comprehensive representation of the 3D spatial partitioning induced by intersecting triangulated surfaces. The work also encompasses efficient mesh co-refinement and mesh repair operations necessary for achieving reliable geometry processing.

Methodology and Key Contributions

The proposed algorithm revolutionizes several aspects of geometric computations in mesh-based CSG:

1. Exact Intersection Algorithm: The core contribution lies in developing an exact mesh intersection technique that accurately computes the Weiler model. The Weiler model delineates how 3D space is divided into volumetric regions by intersecting surfaces. This approach ensures precision in Boolean operations across multiple operands.
2. Constrained Delaunay Triangulation (CDT): The paper introduces a method for performing CDT within intersected mesh triangles—a key process that enhances the robustness of the resultant triangulation by ensuring triangles have optimal quality and topological consistency, even in the presence of intersecting planes.
3. Geometric Kernels: Two geometric kernels are implemented for exact predicates and constructions:
   • Arithmetic expansions, which facilitate handling the arithmetic intricacies underlying geometric predicates.
   • Multi-precision floating-point arithmetic to supplement cases where arithmetic expansions encounter overflow or underflow issues.
4. Practical Implementation: The method includes considerations for a heavy-load intersection scenario managed through a global vertex table and meaningful optimizations like compression strategies for expansions. These make the approach computationally feasible for large-scale meshes.
5. Application and Evaluation: Using datasets such as the Thingi10K and a specially curated ThingiCSG set, the algorithm was extensively tested against conventional and advanced CSG processing pipelines. Results indicate advanced robustness and accuracy, with performance comparisons showcasing competitive or superior timings versus existing methods.

Implications and Future Directions

The implications of this work are profound in domains requiring precise mesh processing, such as CAD, simulation, and scientific visualization. The exact computation approach eliminates errors induced by floating-point arithmetic, thereby enhancing the reliability of downstream tasks such as finite element analysis or animations where precise contacts between parts significantly influence outcomes.

Moving forward, potential areas of development include:

• Snap Rounding Integration: While the work has laid a robust foundational framework, a future direction involves integrating effective snap rounding techniques to convert exact representations back to floating-point while retaining topological properties.
• Performance Improvements: Exploring parallelization strategies or integrating GPU acceleration, particularly leveraging SIMD instructions, could further reduce computation times, making the process more scalable.

The approach also opens a dialogue for further optimization of CDT algorithms in handling constraints with high computational complexity while maintaining precision in results, potentially contributing to both the theoretical and applied facets of geometric computing. The contrast between predicate-oriented error handling and direct geometric constructions emerges as a domain for further exploration, particularly in optimizing 3D modeling engines. The intersections of exact arithmetic with heuristic-based floating-point strategies offer a fertile ground for investigation.