Pythagorean hodograph methods for curves and surfaces

The methods of Computer Aided Geometrie Design (CAGD) form the mathematical basis of the CAD (Computer Aided Design) and CAM (Computer Aided Manufacturing) technologies which are now used almost everywhere in industry. As a relatively new field, CAGD experiences a dynamic expansion characterized by the interaction and exploitation of ideas from various other research domains. The future development of CAD/CAM will benefit from the use of more intelligent representations for geometrical objects. In the proposed project we plan to study one of them, the so-called Pythagorean hodograph (PH) curves and surfaces. The use of these objects provides elegant solutions to various difficult problems occurring in applications in particular in the context of CNC (computer-numerical-control) machining. In the ferst part of this project, the computational techniques for the generation of classical PH curves will be further developed and completed, starting from existing results. More precisely, we plan to develop new constructions for PH spline curves, and to provide new results conceming their behaviour (e.g., conceming the approximation order). Next, we will explore PH curves in the Minkowski spaces of dimension 3 and 4. Recently, these curves were shown to be important because of their close relation to the so-called medial axis transform. So far, virtually no constructions of PH curves in Minkowski space are available. We intend to develop new methods for these settings. The planned project also includes two completely new ideas. First, we plan to discuss the implicit counterparts of Pythagorean hodograph curves and surfaces. Second, we plan to study the new class of "almost PH" curves and surfaces. The topic of the project is not only relevant for various problems from industry, but it is also interesting from the theoretical point of view. It is related to a wide range of disciplines from mathematics and computer science, including differential and algebraic geometry, numerical analysis, approximation theory and Clifford algebra. We will have to use suitable methods from all fields, as well as computer algebra.