Sven Schönherr

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

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.

An efficient, exact, and generic quadratic programming solver for geometric optimization

We present a solver for quadratic programming problems, which is tuned for applications in computational geometry. The solver implements a generalization of the simplex method to quadratic programs. Unlike existing solvers, it is e cient if the problem is dense and has few variables or few constraints. The range of applications covers well-known problems like smallest enclosing ball, or polytope distance, but also linear programming problems like smallest enclosing annulus. We provide an exact implementation with only little overhead compared to pure oating-point code. Moreover, unlike all methods for these problems that were suggested (and implemented) before in computational geometry, the runtime in practice is not exponential in the dimension of the problem, which for example allows to compute smallest enclosing balls in dimensions up to 300 (beyond that, the exact arithmetic becomes the limiting factor). The solver follows the generic programming paradigm, and it will become part of the European computational geometry algorithms library Cgal.

Exact primitives for smallest enclosing ellipses

The problem of finding the unique closed ellipsoid of smallest volume enclosing an n-point set P in d-space (known as the Loroner-John ellipsoid of P [5]) is an instance of convex programming and can be solved by general methods in time O(n) if the dimension is tixed [12, 6, 3, 1]. The problernspeciflc parts of these methods are encapsulated in primitiu e operations that deal with subproblems of constant sise. We derive explicit formulae for the primitive operations of Welsl’s randomised method [12] in dimension d = 2. Compared to previous ones [9, 7, 8], these formulae me simpler and faster to evaluate, and they only contain rational expressions, zdlowing for an exact solution.

Exact primitives for smallest enclosing ellipes

Abstract The problem of finding the unique closed ellipsoid of smallest volume enclosing an n -point set P in d -space (known as the Lowner-John ellipsoid of P (John, 1948)) is an instance of convex programming and can be solved by general methods in time O( n ) if the dimension is fixed (Welzl, 1991; Matousek et al., 1992; Dyer, 1992; Adler and Shamir, 1993). The problem-specific parts of these methods are encapsulated in primitive operations that deal with subproblems of constant size. We derive explicit formulae for the primitive operations of Welzls randomized method (Welzl, 1991) in dimension d = 2. Compared to previous ones (Silverman and Titterington, 1980; Post, 1982; Schonherr, 1994), these formulae are simpler and faster to evaluate, and they only contain rational expressions, allowing for an exact solution. rights reserved.

Ein neues algorithmisches Verfahren zur Fluoroskopie-basierten Neuronavigation

Bei neurochirurgischen Eingriffen an der Wirbelsaule wird die Fluoroskopie als bildgebendes Verfahren eingesetzt, um die raumliche Lage von Instrumenten und Implantaten zu erkennen und gegebenenfalls zu korrigieren. Die haufige Wiederholung solcher Aufnahmen hat eine Reihe von Nachteilen fur Patient und Operateur. Zur Vermeidung dieser Probleme wird eine Technik zur virtuellen Navigation vorgestellt, die es in Kombination mit einem Trackingsystem erlaubt, die Lage von Instrumenten in vorher aufgenommene Fluoroskopiebilder zu projizieren und diese dem Operateur auf einem Bildschirm anzuzeigen.