Appeared in Computational Geometry -- Theory and Applications 13, 1999, 65-90. 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. Key words: Library design; Generic programming; Combinatorial data structure; Polyhedral surface; Halfedge data structure 1 Introduction Combinatorial structures, such as planar maps, are fundamental in computational geometry. In order to be useful in practice, a solid library for compu...

this paper has been supported in part by ESPRIT IV LTR Projects No. 21957 (CGAL) and No. 28155 (GALIA), by the USA-Israel Binational Science Foundation, by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities (Center for Geometric Computing and its Applications), by a Franco-Israeli research grant "factory of the future" (monitored by AFIRST/France and The Israeli Ministry of Science), and by the Hermann Minkowski -- Minerva Center for Geometry at Tel Aviv University

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We show that the combination of the Cgal framework for geometric computation and the number type leda_real yields easy-to-write, correct and efficient geometric programs.

In this paper we describe and discuss a new kernel design for geometric computation in the plane. It combines different kinds of floating-point filter techniques and a lazy evaluation scheme with the exact number types provided by LEDA allowing for efficient and exact computation with rational and algebraic geometric objects. It is the first kernel design which uses floating-point filter techniques on the level of geometric constructions. The experiments we present -- partly using the CGAL framework -- show a great improvement in speed and -- maybe even more important for practical applications -- memory consumption when dealing with more complex geometric computations. 1 Introduction Geometric algorithms are usually designed for the so-called Real RAM, a random access machine that can handle real numbers at unit cost. The exact computation paradigm ([16]) advocates to give the implementer of a geometric algorithm the illusion of a real RAM by providing exact number types and exact ...

We report on the use of the generic programming paradigm in the computational geometry algorithms library cgal. The parameterization of the geometric algorithms in cgal enhances exibility and adaptability and opens an easy way for abolishing precision and robustness problems by exact but nevertheless ecient computation. Furthermore we discuss circulators, which are an extension of the iterator concept to circular structures. Such structures arise frequently in geometric computing. 1 Introduction cgal is a C++ library of geometric algorithms and data structures. It is developed by several sites in Europe and Israel. The goal is to enhance the technology transfer of the algorithmic knowledge developed in the eld of computational geometry to applications in industry and academia. Computational geometry is the sub-area of algorithm design that deals with the design and analysis of algorithms for geometric problems involving objects like points, lines, polygons, and polyhedra. Ove...

Contents Acknowledgments iii 1 Introduction 3 2 Preliminaries and Related Work 9 2.1 CGAL and Generic Programming . . . . . . . . . . . . . . . . 9 2.2 Robustness in Geometric Computation . . . . . . . . . . . . . 11 2.2.1 Exact Arithmetic . . . . . . . . . . . . . . . . . . . . . 12 2.2.2 Floating Point Filters . . . . . . . . . . . . . . . . . . . 13 2.3 Planar Maps in CGAL . . . . . . . . . . . . . . . . . . . . . . 16 2.3.1 Geometric Traits . . . . . . . . . . . . . . . . . . . . . 16 2.3.2 Point Location Strategies . . . . . . . . . . . . . . . . . 17 2.4 Computing the Combinatorial Structure of an Arrangement . 18 2.4.1 Conditions for Curves . . . . . . . . . . . . . . . . . . 18 2.4.2 Isolation of Intersection Points . . . . . . . . . . . . . . 21 3 Arrangements in CGAL 23 3.1 Hierarchy Tree . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Design and Implementation Details . . . . . . . . . . . . . . . 26 3.2.1 General Design . . . . . . .

We introduce a new filtering technique that can be used in the implementation of geometric algorithms called "structural filtering". Using this filtering techniques we gain about 20 % when compared to predicate-filtered implementations. Of theoretical interest are some results regarding the robustness of sorting algorithms against erroneous comparisons.

The number type leda real provides exact computation for a subset of real algebraic numbers: Every integer is a leda real, and leda reals are closed under the basic arithmetic operations +, −, ∗, / and k-th root operations. leda reals guarantee correct results in all comparison operations. The number type is available

The problem of computing the convex hull of a finite set of points arises often in practice and is a cornerstone of any geometric software library. The 2- and 3-dimensional cases are of special interest due to their direct correspondence with physical entities. Several algorithms have been designed and developped for general dimension but also for this case. The aim of this work is to apply existing implementations of general as well as specific algorithms, namely quickhull, including those from general libraries such as Cgal and Leda, on real and artificially constructed problems and compare their performance in terms of speed, accuracy, and robustness. 1 Introduction The problem of computing the convex hull of a set of points V = fv 1 ; v 2 ; : : : ; v ng in a d-dimensional space, that is computing the smallest convex set containing V , has been intensively studied during the last thirty years. It is a problem of complexity \Omega\Gamma n log n + n bd=2c ), for fixed d. This numb...