Boolean operations on 3D selective Nef complexes: Data structure, algorithms, optimized implementation and experiments

Abstract

Nef polyhedra in d-dimensional space are the closure of half-spaces under boolean set operations. In consequence, they can represent non-manifold situations, open and closed sets, mixed-dimensional complexes, and they are closed under all boolean and topological operations, such as complement and boundary. They were introduced by W. Nef in his seminal 1978 book on polyhedra. The generality of Nef complexes is essential for some applications.In this paper, we present a new data structure for the boundary representation of three-dimensional Nef polyhedra and efficient algorithms for boolean operations. We use exact arithmetic to avoid well-known problems with floating-point arithmetic and handle all degeneracies. Furthermore, we present important optimizations for the algorithms, and evaluate this optimized implementation with extensive experiments. The experiments supplement the theoretical runtime analysis and illustrate the effectiveness of our optimizations. We compare our implementation with the ACIS CAD kernel. ACIS is mostly faster, by a factor up to six. There are examples on which ACIS fails.The implementation was released as Open Source in the Computational Geometry Algorithm Library (CGAL) release 3.1 in December 2004.

Processing BIM and GIS Models in Practice: Experiences and Recommendations from a GeoBIM Project in The Netherlands

“…An attempt was made to parse and process every object in an IFC file independently using a modified version of IfcOpenShell, where its Open CASCADE (https://www.opencascade.com) -based kernel has been substituted for a new one based on CGAL. The aim of this approach was to use spatial analysis algorithms from the GIS domain and the robust Boolean set operations on Nef polyhedra available in CGAL in order to solve various use cases [29][30][31]. This approach was chosen since previous experience showed that the Boolean set operations in Open CASCADE are not as robust as those available in CGAL.…”

Implicit Real Vector Automata

“…There can be edges, facets, volumes and some more information to link those elements with respect to the incidence between them. This later information is useful to guarantee a certain efficiency of the manipulation algorithms, although the point decision problem is quadratic in the size of the polyhedron [Hac07].…”

A Practical and Robust Method to Compute the Boundary of Three-dimensional Axis-aligned Boxes

“…Most of them classify the set of CSG primitives with some kind of spatial decomposition scheme (Thibault and Naylor, 1987). Hachenberger et al have studied and optimized boolean operations on Nef Polyhedra, and that handle all cases, including all degeneracies (Hachenberger and Kettner, 2005). Baumann et al compute the union of three dimensional axis-aligned boxes, represented by a set of enclosing rectangles, by an octree-based incremental algorithm (Baumann et al, 2008).…”