On the square-freeness of the offset equation to a rational planar curve

TL;DR

This paper provides an algebraic characterization of the multiplicity of factors in the offset equation of a rational planar curve, establishing conditions under which the resultant is square-free and offering a test for offset curves.

Contribution

It completes the algebraic description of the resultant for offset curves by determining factor multiplicities and proving when the resultant is square-free for proper parametrizations.

Findings

Factors corresponding to simple components have multiplicity 1

Special components, if any, have multiplicity 2

The non-extraneous part of the resultant is square-free under certain conditions

Abstract

It is well known that an implicit equation of the offset to a rational planar curve can be computed by removing the extraneous components of the resultant of two certain polynomials computed from the parametrization of the curve. Furthermore, it is also well known that the implicit equation provided by the non-extraneous component of this resultant has at most two irreducible factors. In this paper, we complete the algebraic description of this resultant by showing that the multiplicity of the factors corresponding to the offset can be computed in advance. In particular, when the parametrization is proper, i.e. when the curve is just traced once by the parametrization, we prove that any factor corresponding to a simple component of the offset has multiplicity 1, while the factor corresponding to the special component, if any, has multiplicity 2. Hence, if the parametrization is proper and there is no special component, the non-extraneous part of the resultant is square-free. In fact, this condition is proven to be also sufficient. Additionally, this result provides a simple test to check whether or not a given rational curve is the offset of another curve.