Computational Line Geometrie

Line geometry investigates the set of lines in 3-space. There is a rich literature on this branch of classical geometry. It possesses a close relation to mechanics and to spatial kinematics and has therefore found applications in mechanism design and robot kinematics. Other applications include motion planning, placement and assembly problems, object recognition in computer vision, reverse engineering of geometric models, surface design and wire electrical discharge machining. Despite this broad area of applications, a careful study of line geometry from the computational point of view has not been undertaken so far. It is the purpose of the present research project to lay the fundamentals of this rather unexplored field of geometry, which could be called computational line geometry. A central part of our work will be the development of approximation and robust regression techniques in line space. It has to be combined with the introduction of appropriate metrics in line space, which focus on the domain of interest in which the line is realized in a physical object. This is essential for reliable practical application of line geometric results. Other topics of the project are algorithms for line congruences in connection with an application in 5-axis sculputured surface machining, algorithms for ruled surfaces and visualization techniques for the Klein model of line geometry. The latter topic arises from our work on a monograph on computational line geometry. The main purpose of the project is to lay the mathematical fundamentals of computational line geometry. The efficiency of the algorithms will be illustrated at hand of problems from specific application areas. Software development is planned for the application in NC machining. This is part of an international research cooperation whose goal is the development of software for collision free 5-axis machining of sculptured surfaces, which shall be, portable to major CAD/CAM systems.

Within this project we developed the mathematical fundamentals of a computer-aided treatment of problems, in which the geometry of lines in three-dimensional space is playing a central role. Line geometry studies the set of lines in three-dimensional space. It is a well understood area of classical geometry and it possesses close relations to mechanics and space kinematics. Classical applications are in robotics and mechanism design. New applications of line geometry are in motion planning, positioning and manufacturing problems, recognition of objects in Computer Vision, object reconstruction from scanner data for reverse engineering of geometric objects, and in wire cut EDM machining. All these applications involve line geometry from the geometric computing point of view. The study of the mathematical fundamentals of this area, coined `Computational Line Geometry`, has been the focus of the present project. Special emphasis has been on approximation problems in line space. More generally, we dealt with approximation in spaces of geometric objects. Such problems occur in the reconstruction of objects from clouds of measurement points (e.g., laser scanner data) and in the stability analysis of positions of parallel robots. We also encountered applications of computational line geometry to the analysis of tolerances in CAD constructions. Moreover, the present project possesses a close relation to motion planning for five-axis machining of sculptured surfaces. The project results, along with a description of the essential classical material, have been published in a book `Computational Line Geometry`. It appeared in 2001 in the new series `Mathematics and Visualization` of Springer- Verlag.