Nef Polyhedra: A Brief Introduction

Bieri, Hanspeter

doi:10.1007/978-3-7091-7584-2_3

Cited by 20 publications

(17 citation statements)

References 21 publications

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“…The simplification is based on Nef's theory [Nef78,Bie95] that provides a straightforward classification of point neighborhoods; the simplification just eliminates those neighborhoods that cannot occur in Nef polyhedra. The merge operation of regions in step (i) uses a union find data structure [CLR90] to efficiently update the pointers in the half-edge data structure associated with the regions.…”

Section: Construction Of a Sphere Mapmentioning

confidence: 99%

Boolean Operations on 3D Selective Nef Complexes: Data Structure, Algorithms, and Implementation

Granados

Hachenberger

Hert

et al. 2003

Algorithms - ESA 2003

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Section: Construction Of a Sphere Mapmentioning

confidence: 99%

Boolean Operations on 3D Selective Nef Complexes: Data Structure, Algorithms, and Implementation

Granados

Hachenberger

Hert

et al. 2003

Algorithms - ESA 2003

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“…The simplification is based on Nef's theory [7,36] that provides a straightforward classification of point neighborhoods; the simplification just eliminates those neighborhoods that cannot occur in Nef polyhedra. The merge operation of regions in step (i) uses a union find data structure [11] to keep track of merged regions.…”

Section: Construction Of a Sphere Mapmentioning

confidence: 99%

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

Hachenberger

Kettner

Mehlhorn

2007

Computational Geometry

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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.

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“…The paper is structured as follows: We omit the underlying theory of Nef polyhedra and refer to [Nef 1978;Bieri 1995] and our previous paper [Granados et al 2003]. We summarize our data structure in Section 2 and our algorithms for generalized set operations in Section 3.…”

Section: Structurementioning

confidence: 99%

“…The simplification is based on Nef's theory [Nef 1978;Bieri 1995] that provides a straightforward classification of point neighborhoods; the simplification just eliminates those neighborhoods that cannot occur in Nef polyhedra. The merge operation of regions in step (i) uses a union find data structure [Cormen et al 1990] to efficiently update the pointers in the half-edge data structure associated with the regions.…”

Section: Construction Of a Sphere Mapmentioning

confidence: 99%

Boolean operations on 3D selective Nef complexes

Hachenberger

Kettner

2005

Proceedings of the 2005 ACM Symposium on Solid and Physical Modeling

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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 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 pointlocation 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 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 CGAL release 3.1 in December 2004.

show abstract

Nef Polyhedra: A Brief Introduction

Cited by 20 publications

References 21 publications

Boolean Operations on 3D Selective Nef Complexes: Data Structure, Algorithms, and Implementation

Boolean Operations on 3D Selective Nef Complexes: Data Structure, Algorithms, and Implementation

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

Boolean operations on 3D selective Nef complexes

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