FWF - Computational geometry - NFN Industrial Geometry

Project: Research project

Description

Industrial Geometry - An emerging research area




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




Subproject S09205 (Computational Geometry)



(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.





Subproject S09209 (Computational Differential Geometry)




(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.

StatusFinished
Effective start/end date1/04/0531/12/11