Programming with CGAL

2007

We present a C++ open-source implementation of an incremental algorithm for the computation of the Voronoi diagram of ellipses in the Euclidean plane. This is the first complete implementation, under the exact computation paradigm, for the given problem. It is based on the cgal package for the Apollonius diagram in the plane: exploiting the generic programming principle, our main additions concern the predicates. The ellipses are input in parametric representation, which allows us to develop a GUI for input and output. The software concerns non-intersecting ellipses, but the extension to the general case is possible.

Arrangements of planar curves are fundamental structures in computational geometry. Recently, the arrangement package of Cgal, the Computational Geometry Algorithms Library, has been redesigned and re-implemented exploiting several advanced programming techniques. The resulting software package, which constructs and maintains planar arrangements, is easier to use, to extend, and to adapt to a variety of applications, is more efficient space-and timewise, and is more robust. The implementation is complete in the sense that it handles degenerate input, and it produces exact results. In this paper we describe how various programming techniques were used to accomplish specific tasks within the context of Computational Geometry in general and Arrangements in particular. A large set of benchmarks assured the successful applications of the adverted programming techniques. The results of a small sample are reported at the end of this article.

We present a basic data structure for geometric data which can be adapted to represent common geometry representations like CSG, BSP, aso. The new data structure has been designed to be easy to use, and easy to extend. Due to the representation of geometric data using a directed acyclic graph, a number of the standard rendering algorithms can be used on the data structure in a very straightforward way. The new data structure has been implemented as a C++ library and can therefore serve as high-level tool for developing graphics applications, or as an extension for using C++ as a modeling language.

Discrete Mathematics and Its Applications, 2004

Over the past decades, two major software libraries that support a wide range of geometric computing have been developed: Leda, the Library of Efficient Data Types and Algorithms, and Cgal, the Computational Geometry Algorithms Library. We start with an introduction of common aspects of both libraries and major differences. We continue with sections that describe each library in detail. Both libraries are written in C++. Leda is based on the object-oriented paradigm and Cgal is based on the generic programming paradigm. They provide a collection of flexible, efficient, and correct software components for computational geometry. Users should be able to easily include existing functionality into their programs. Additionally, both libraries have been designed to serve as platforms for the implementation of new algorithms. Correctness is of crucial importance for a library, even more so in the case of geometric algorithms where correctness is harder to achieve than in other areas of software construction. Two well-known reasons are the exact arithmetic assumption and the nondegeneracy assumption that are often used in geometric algorithms. However, both assumptions usually do not hold: floating-point arithmetic is not exact and inputs are frequently degenerate. See Chapter 45 for details. EXACT ARITHMETIC There are basically two scientific approaches to the exact arithmetic problem. One can either design new algorithms that can cope with inexact arithmetic or one can use exact arithmetic. Instead of requiring the arithmetic itself to be exact, one can guarantee correct computations if the so-called geometric primitives are exact. So, for instance, the predicate for testing whether three points are collinear must always give the right answer. Such an exact primitive can be efficiently implemented using floating-point filters or lazy evaluation techniques. This approach is known as the exact geometric computing paradigm and both libraries, Leda and Cgal, adhere to this paradigm. However, they also offer straight floating-point implementations. DEGENERACY HANDLING An elegant (theoretical) approach to the degeneracy problem is symbolic perturbation. But this method of forcing input data into general position can cause serious problems in practice. In many cases, it increases the complexity of (interme-1799

Computational Geometry, 1998

In this paper we describe and discuss a kernel for higher-dimensional computational geometry and we present its application in the calculation of convex hulls and Delaunay triangulations. The kernel is available in form of a software library module programmed in C++ extending LEDA. We introduce the basic data types like points, vectors, directions, hyperplanes, segments, rays, lines, spheres, affine transformations, and operations connecting these types. The description consists of a motivation for the basic class layout as well as topics like layered software design, runtime correctness via checking routines and documentation issues. Finally we shortly describe the usage of the kernel in the application domain.

Southeast Europe Journal of Soft Computing

Computational geometry is an integral part of mathematics and computer science deals with the algorithmic solution of geometry problems. From the beginning to today, computer geometry links different areas of science and techniques, such as the theory of algorithms, combinatorial and Euclidean geometry, but including data structures and optimization. Today, computational geometry has a great deal of application in computer graphics, geometric modeling, computer vision, and geodesic path, motion planning and parallel computing. The complex calculations and theories in the field of geometry are long time studied and developed, but from the aspect of application in modern information technologies they still are in the beginning. In this research is given the applications of computational geometry in polygon triangulation, manufacturing of objects with molds, point location, and robot motion planning.

2008

construction of a Straight Skeleton on the interior of a 2D Polygon with Holes.

CGALmesh is the mesh generation software package of the Computational Geometry Algorithm Library (CGAL). It generates isotropic simplicial meshes-surface triangular meshes or volume tetrahedral meshes-from input surfaces, 3D domains as well as 3D multi-domains, with or without sharp features. The underlying meshing algorithm relies on restricted Delaunay triangulations to approximate domains and surfaces, and on Delaunay refinement to ensure both approximation accuracy and mesh quality. CGALmesh provides guarantees on approximation quality as well as on the size and shape of the mesh elements. It provides four optional mesh optimization algorithms to further improve the mesh quality. A distinctive property of CGALmesh is its high flexibility with respect to the input domain representation. Such a flexibility is achieved through a careful software design, gathering into a single abstract concept, denoted by the oracle, all required interface features between the meshing engine and the input domain. We already provide oracles for domains defined by polyhedral and implicit surfaces.