The implicit equation of a canal surface

Marc Dohm 1, 2 Severinas Zube 3
2 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : A canal surface is an envelope of a one parameter family of spheres. In this paper we present an efficient algorithm for computing the implicit equation of a canal surface generated by a rational family of spheres. By using Laguerre and Lie geometries, we relate the equation of the canal surface to the equation of a dual variety of a certain curve in 5-dimensional projective space. We define the μ-basis for arbitrary dimension and give a simple algorithm for its computation. This is then applied to the dual variety, which allows us to deduce the implicit equations of the the dual variety, the canal surface and any offset to the canal surface.
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Marc Dohm, Severinas Zube. The implicit equation of a canal surface. Journal of Symbolic Computation, Elsevier, 2009, 44 (2), pp.111-130. ⟨10.1016/j.jsc.2008.06.001⟩. ⟨hal-00290577⟩

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