Computer Aided Geometric Design (CAGD) and Visualization play an important role in the construction of such varied products as cars, ships, airplanes etc. A practical implementation has to balance two conflicting goals, namely efficiency and reliability. Typically, a numerical algorithm is efficient, but it is not reliable and may fail in certain cases, especially in areas near the singularities. On the other hand, algorithms based on exact arithmetic are reliable, but are normally slow and require a lot of memory space. Perhaps the goals of efficiency and reliability cannot be met simultaneously without some compromises.
We propose to negotiate those compromises judiciously. That means that, we will tackle the problems by a hybrid symbolic-numerical approach and develop algorithms which should fit those three main goals.
Together with the theoretical research, we also propose a project work which is directed towards designing software. The software - called HySaX (Hybrid Software for Algebraic Geometry in X-windows environment) - will aim at the implementation of hybrid symbolic-numerical algorithms for approximation and representation of algebraic sets in two and three dimensions. The software will be built as an enhancement of the language Maple, a traditional and widely used computer algebra system. Therefore it can be benefited from a large number of already known algorithms which has been implemented in the library/share-library of Maple. Especially, it will succeed the system CASA, a computer algebra system for constructive algebraic geometry, which has been developed by the Computer Algebra group of RISC-Linz.