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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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Problems of Geometric Computing arise in a variety of research
areas including Computer Aided Design, Geometric Modeling, Computer Vision
and Robotics. Recent research demonstrated that classical geometric
methods and appropriate extensions of those can greatly simplify
the solution of problems and the development of algorithms in geometric
computing.

In the present tutorial we will focus on classical geometric tools,
mainly from sphere and line geometry, applied to the symbolic computation
of parameterizations for algebraic curves and surfaces. In particular,
we will provide an introduction to classical Laguerre sphere geometry
and its various models. It will be shown how the change from
the standard model to the so-called isotropic model offers a simple
and elegant derivation of all rational curves and surfaces with
rational offsets. The problem of determining these curves and surfaces
has been posed by Farouki and Sakkalis in their seminal paper on
Pythagorean hodograph curves. Moreover, the computation of
rational parameterizations for certain envelope surfaces (pipe surfaces,
canal surfaces, offsets of ruled surfaces and quadrics) will be derived
with Laguerre sphere geometry. These surfaces have various applications
in Computer Aided Design and geometric modeling.

In generalization of offsets, we are also treating Minkowski sums
of surfaces and derive rational parameterizations for certain classes
of Minkowski sum boundary surfaces (convolution surfaces). This
material is related to research on geometric tolerancing and error
propagation in solid modeling systems.

The focus of this tutorial is on the use of classical geometric methods.
Basically they serve in this area as a preprocessing and simplification
step for systems of polynomial equations which are hardly solvable in 
a direct way by current computer algebra systems. 


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