Rational Parametrizations for Envelopes of Natural
Quadrics
Date: July 18th (Thursday)
Time: 09:30-10:00
Abstract
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.
