Lutz Kettner

On the design of CGAL a computational geometry algorithms library

CGAL is a Computational Geometry Algorithms Library written in C++, which is being developed by research groups in Europe and Israel. The goal is to make the large body of geometric algorithms developed in the field of computational geometry available for industrial application. We discuss the major design goals for CGAL, which are correctness, flexibility, ease‐of‐use, efficiency, and robustness, and present our approach to reach these goals. Generic programming using templates in C++ plays a central role in the architecture of CGAL. We give a short introduction to generic programming in C++, compare it to the object‐oriented programming paradigm, and present examples where both paradigms are used effectively in CGAL. Moreover, we give an overview of the current structure of the CGAL‐library and consider software engineering aspects in the CGAL‐project. Copyright

Using generic programming for designing a data structure for polyhedral surfaces

Abstract Software design solutions are presented for combinatorial data structures, such as polyhedral surfaces and planar maps, tailored for program libraries in computational geometry. Design issues considered are flexibility, time and space efficiency, and ease-of-use. We focus on topological aspects of polyhedral surfaces and evaluate edge-based representations with respect to our design goals. A design for polyhedral surfaces in a halfedge data structure is developed following the generic programming paradigm known from the Standard Template Library STL for C++. Connections are shown to planar maps and face-based structures.

Classroom examples of robustness problems in geometric computations

The algorithms of computational geometry are designed for a ma- chine model with exact real arithmetic. Substituting floating point arithmetic for the assumed real arithmetic may cause implementations to fail. Although this is well known, there is no comprehensive documentation of what can go wrong and why. In this extended abstract, we study a simple incremental algorithm for planar convex hulls and give examples which make the algorithm fail in all pos- sible ways. We also show how to construct failure-examples semi-systematically and discuss the geometry of the floating point implementation of the orientation predicate. We hope that our work will be useful for teaching computational ge- ometry. The full paper is available at www.mpi-sb.mpg.de/˜mehlhorn/ ftp/ClassRoomExamples.ps. It contains further examples, more theory, and color pictures. We strongly recommend to read the full paper instead of this extended abstract.

The CGAL Kernel: A Basis for Geometric Computation

A large part of the Cgal-project is devoted to the development of a Computational Geometry Algorithms Library, written in C++. We discuss design issues concerning the Cgal-kernel which is the basis for the library and hence for all geometric computation in Cgal.

Designing a data structure for polyhedral surfaces

Design solutions for a program library are presented for combinatorial data structures in computational geometry, such as planar maps and polyhedral surfaces. Design issues considered are genericity, flcsibility, time and space efficiency, and ease-of-use. We focus on topological aspects of polyhedral surfaces. Edge-based reprew%ations for polyhedrons are evaluated with respect to the design goals. A design for polyhedral surfaces in a halfedge data structure is developed following the generic programming paradigm known from the Standard Template Library STL for C++. Connections arc shown to planar maps and face-based structures managing holes in facets.

Boolean operations on 3D selective Nef complexes : data structure, algorithms, and implementation

We describe a data structure for three-dimensional Nef complexes, algorithms for boolean operations on them, and our implementation of data structure and algorithms. Nef polyhedra were introduced by W. Nef in his seminal 1978 book on polyhedra. They are the closure of half-spaces under boolean operations and can represent non-manifold situations, open and closed boundaries, and mixed dimensional complexes. Our focus lies on the generality of the data structure, the completeness of the algorithms, and the exactness and efficiency of the implementation. In particular, all degeneracies are handled.

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

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.

An exact, complete and efficient implementation for computing planar maps of quadric intersection curves

We present the first exact, complete and efficient implementation that computes for a given set P=p1,...,pn of quadric surfaces the planar map induced by all intersection curves p1∩ pi, 2 ≤ i ≤ n, running on the surface of p1. The vertices in this graph are the singular and x-extreme points of the curves as well as all intersection points of pairs of curves. Two vertices are connected by an edge if the underlying points are connected by a branch of one of the curves. Our work is based on and extends ideas developed in [20] and [9].Our implementation is complete in the sense that it can handle all kind of inputs including all degenerate ones where intersection curves have singularities or pairs of curves intersect with high multiplicity. It is exact in that it always computes the mathematical correct result. It is efficient measured in running times.

Boolean operations on 3D selective Nef complexes: optimized implementation and experiments

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 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 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.

Tight degree bounds for pseudo-triangulations of points

We show that every set of n points in general position has a minimum pseudo-triangulation whose maximum vertex degree is five. In addition, we demonstrate that every point set in general position has a minimum pseudotriangulation whose maximum face degree is four (i.e., each interior face of this pseudo-triangulation has at most four vertices). Both degree bounds are tight. Minimum pseudo-triangulations realizing these bounds (individually but not jointly) can be constructed in O(n logn) time.