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Abstract

Nef polyhedra in $d$-dimensional space are the closure of half-spaces under boolean set operation. 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. We presented in previous work a new data structure for the boundary representation of three-dimensional Nef polyhedra with efficient algorithms for boolean operations. These algorithms were designed for correctness and can handle all cases, in particular all \emph{degeneracies}. To this extent we rely on exact arithmetic to avoid well known problems with floating-point arithmetic. In this paper, we present important optimizations for the algorithms. We describe the chosen implementations for the point-location and the intersection-finding subroutines, a kd-tree and a fast box-intersection algorithm, respectively. We evaluate this optimized implementation with extensive experiments that supplement the runtime analysis from our previous paper and that illustrate the effectiveness of our optimizations. We compare our implementation with the \textsc{Acis} CAD kernel and demonstrate the power and cost of the exact arithmetic in near-degenerate situations. The implementation was released as Open Source in the \textsc{Cgal} release 3.1 in December 2004.