The implicit equation of a canal surface

TL;DR

This paper introduces an efficient algorithm to compute the implicit equation of canal surfaces generated by rational sphere families, utilizing Laguerre and Lie geometries and dual varieties in projective space.

Contribution

It presents a novel method linking canal surface equations to dual varieties using bc-bases and geometric transformations, enabling explicit implicit equation computation.

Findings

Algorithm computes implicit equations efficiently.

Relates canal surfaces to dual varieties in 5D projective space.

Provides a method for offsets of canal surfaces.

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 \mu-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.