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Rational Parametrizations for Envelopes of Natural Quadrics
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Helmut Pottmann, Martin Peternell
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Date:  July 18th (Thursday)\\
Time:  09:30-10:00\\

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Abstract
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We define real rational one-parameter sets of natural quadrics (spheres, cones and cylinders of
revolution) in Euclidean 3-space and prove that the envelope of such a surface family is a rational
surface with rational offsets. The proof is constructive and is based on ideas from classical
Laguerre geometry. In particular this result possesses the following quite surprising corollaries.
Rational canal surfaces with rational spine curve, in particular rational pipe surfaces with rational
spine curve are rational. Rational non-developable ruled surfaces possess rational offsets. The
offsets of regular, i.e. non-developable quadrics can be rationally parametrized. Particular
emphasis is laid onto techniques for constructing low degree parametrizations. 

