While current CAD-systems support piecewise - generally rational - parametric patches, they only support algebraic surfaces of degree 1 and 2. Consequently the introduction of real algebraic surfaces in industrial value chains using CAD-represented geometry calls for efficient algorithms for computing rational parametric curve and surface representations of real algebraic curves and surfaces. Parameterizing general algebraic surfaces is difficult and challenging. For algebraic curves and surfaces of degree larger than 2 frequently no rational parameterizations exist. Consequently computational techniques for approximate rational parameterization of curves and surfaces are necessities for a smooth introduction of algebraic surfaces in CAD-geometry industrial value chains. Until now such parameterization techniques have received little attention.

The project will both address the challenges of approximate parameterization and more theoretical questions related to so-called universal parameterizations. These are rational mappings that allow generating rational patches on certain special classes of algebraic surfaces. They are related to representation formulas of the solutions to the corresponding equation in unique factorization domains. Recently similar formulas, which are valid in the subclass of principal ideal domains, have emerged. These results need to be analyzed from a geometric point of view, leading to the discussion of related applications.

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