Industrial geometry is based on computational techniques which originated in various areas of applied geometry. To give a few examples, the methods of Computer Aided Geometric Design form the mathematical foundation of the powerful CAD technologies available today. Computer Vision provides methods for inspecting and analyzing images and videos. Image processing is used to reconstruct geometric features from digital image data, such as X-ray or computer tomography images. Computational Geometry provides efficient algorithms for solving fundamental geometrical problems. Robot kinematics deals with geometric problems which occur in connection with robot mechanical systems.
In recent years, these different areas of research have started to become increasingly interconnected, and begun even to merge. A driving force in this process is the increasing complexity of applications, where one field of research alone would be insufficient to achieve useful results. Novel technologies for acquisition and processing of data lead to new and increasingly challenging problems, whose solution requires the combination of techniques from different branches of applied geometry.
Besides Subproject S09201: Coordination and Service, Knowledge transfer and sustainability, TU Graz hosts the following subprojects
(Principal Investigator: Oswin Aichholzer) Computational Geometry is dedicated to the algorithmic study of elementary geometric questions. Traditionally it deals with basic geometric objects like points, lines, and planes. For real world applications, however, often reliable techniques for advanced geometric primitives like surfaces and location query structures are needed.
The role of this project is twofold. On one hand it will provide the theoretical background for advanced geometric algorithms and data structures for several other projects within this joint research project (JRP). These include geometric structures for fast information retrieval, the generation and manipulation of triangular meshes, the computation of suitable distance functions to multidimensional objects, and the representation of advanced geometric objects.
Another aim of this project is to develop novel techniques for the manipulation and optimization of geometric structures. Here the emphasis is on geometric graphs (triangulation-like and Voronoi diagram-like structures, spanning trees). Properties of these structures will be investigated, with the goal of designing more efficient geometric algorithms and data structures.
Existing geometric algorithms libraries (CGAL, LEDA) will be used to guarantee robustness of the developed algorithms.
(Principal investigator: Johannes Wallner) Computational Differential Geometry means methods of both numerical and discrete mathematics with the purpose of investigating and modeling curves and surfaces. The main theme of this research project is the robust analysis of differential properties of surfaces, the creation of discrete and semi-discrete models of freeform surfaces, and the study of geometric properties of such models. It is only recently that the wealth of interesting geometry connected to applications in, say, architecture, has come to the attention of mathematicians, and presumably only a small part of it has been investigated. We are investigating topics of Discrete Differential Geometry: discrete curvatures based on parallel meshes, quad-based and hex-based discrete surfaces, Christoffel duality, and others. New lines of research of semi-discrete surfaces and inverse problems in connection with integral invariants.