ISSAC 2003 Tutorial

Martin Peternell and Helmut Pottmann

Vienna University of Technology

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.