The square-freeness of the offset equation to a rational planar curve, computed via resultants

Abstract

It is well known [Algebraic properties of plane offset curves, Comput. Aided Geom. Des. 7 (1990) 101–127] 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 nonextraneous component of this resultant has at most two irreducible factors [Algebraic analysis of offsets to hypersurfaces, Math. Z. 234 (2000) 697–719]. 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 nonextraneous part of the resultant is square-free. In fact, this condition is proven to be also sufficient. Therefore, this result provides a simple test to check whether or not the offset of a given rational curve has a special component, and in turn, whether a given rational curve is the offset of another curve.