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\centerline{ISSAC 2003 Tutorial}
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\centerline{\bf Classical Geometry for Symbolic Geometric Computing}

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\noindent
\centerline{Martin Peternell and Helmut Pottmann}
\centerline{ Vienna University of Technology}

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

\begin{itemize}

\item  {\it 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?


\item {\it 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,...


\item  {\it 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.

\end{itemize}
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