ISSAC 2003 Tutorial

Classical Geometry for Symbolic Geometric Computing

Martin Peternell and Helmut Pottmann

Vienna University of Technology

Here are some PhD thesis problems to which we introduce in our tutorial.

• Real parameterizations of convolution surfaces from complex ones. The convolution surface of two rational ruled surfaces is a rational surface. It is simple to derive a real rational parameterization. All quadrics in complex 3-space are ruled and thus complex parameterizations of convolution surfaces of quadrics are easily computable. Can we use this fact to derive real rational parameterizations?

• Minkowski sums from a sphere geometric viewpoint. In our tutorial we will show how to use sphere geometry and the cyclographic mapping for the computation of the Minkowski sum of canal surfaces (in particular, surfaces of revolution). In this area, much more could be done: discuss important surface classes, generalize to spheres in other geometries, use sphere geometry not only for the Minkowski sum of canal surfaces but for more general questions of mathematical morphology,...

• Precise tolerance regions in geometric computing. Assuming errors in the input of a geometric computation problem, the question arises how these errors propagate during the algorithm to the output. If the errors in the input are modeled by tolerance regions, it is interesting to describe, as precise as possible, the tolerance regions for the output. Recent research showed how certain affine constructions can nicely be treated in this way. There is a very close relation to Minkowski sums. However, as soon as metric aspects enter the computation, the problem of computing precise tolerance regions becomes harder. We enter here a largely unexplored area of research.