Michael Seel

Checking geometric programs or verification of geometric structures

A program checker verifies that a particular program execution is correct. We give simple and efficient program checkers for some basic geometric tasks. We report about our experiences with program checking in the context of the LEDA system. We discuss program checking for data structures that have to rely on user-provided functions.

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.

Exact geometric predicates using cascaded computation

In this paper we talk about a new efficient numerical approach to deal with inaccuracy when implementing eometric algorithms. Using various floating-point filters together with arbitrary precision packages, we develop an easy-to-use xpression compiler called EXPCOMP. EXPCOMP supports all common operations +Ibines , ‘, /, ,/, Applying a new semi-static filter, EXPCOMP comthe speed of static filters with the power of dynamic filters. The filter stages deal with all kinds of floating-point exceptions, including underflow The resulting programs how a very good runtime behaviour.

An adaptable and extensible geometry kernel

Geometric algorithms are based on geometric objects such as points, lines and circles. The term kernel refers to a collection of representations for constant-size geometric objects and operations on these representations. This paper describes how such a geometry kernel can be designed and implemented in C++, having special emphasis on adaptability, extensibility and efficiency. We achieve these goals following the generic programming paradigm and using templates as our tools. These ideas are realized and tested in Cgal, the Computational Geometry Algorithms Library, see http://www.cgal.org/.

Exact geometric computation using Cascading

In this paper we talk about a new efficient numerical approach to deal with inaccuracy when implementing geometric algorithms. Using various floating-point filters together with arbitrary precision packages, we develop an easy-to-use expression compiler called EXPCOMP. EXPCOMP supports all common operations . Applying a new semi-static filter, EXPCOMP combines the speed of static filters with the power of dynamic filters. The filter stages deal with all kinds of floating-point exceptions, including underflow. The resulting programs show a very good runtime behaviour.

Infimaximal frames: a technique for making lines look like segments

Many geometric algorithms that are usually formulated for points and segments generalize easily to inputs also containing rays and lines. The sweep algorithm for segment intersection is a prototypical example. Implementations of such algorithms do, in general, not extend easily. For example, segment endpoints cause events in sweep line algorithms, but lines have no endpoints. We describe a general technique, which we call infimaximal frames, for extending implementations to inputs also containing rays and lines. The technique can also be used to extend implementations of planar subdivisions to subdivisions with many unbounded faces. We have used the technique successfully in generalizing a sweep algorithm designed for segments to rays and lines and also in an implementation of planar Nef polyhedra.14,1 Our implementation is based on concepts of generic programming in C++ and the geometric data types provided by the C++ Computational Geometry Algorithms Library (CGAL).

Planar Nef Polyhedra and Generic Higher-dimensional Geometry

We present two generic software projects that are part of the software library CGAL. The first part describes the design of a geometry kernel for higher-dimensional Euclidean geometry and the interaction with application programs. We describe the software structure, the interface concepts, and their models that are based on coordinate representation, number types, and memory layout. In the higher-dimensional software kernel the interaction between linear algebra and the geometric objects and primitives is one important facet. In the actual design our users can replace number types, representation types, and the traits classes that inflate kernel functionality into our current application programs: higher-dimensional convex hulls and Delaunay tedrahedralisations. In the second part we present the realization of planar Nef polyhedra. The concept of Nef polyhedra subsumes all kinds of rectilinear polyhedral subdivisions and is therefore of general applicability within a geometric software library. The software is based on the theory of extended points and segments that allows us to reuse classical algorithmic solutions like plane sweep to realize binary operations of Nef polyhedra.

A computational basis for higher-dimensional computational geometry and applications

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.