Implicitization of rational ruled surfaces with μ-bases

Chen, Sederberg, and Zheng introduced the notion of a μ-basis for a rational ruled surface in Chen et al. [Chen, F., Zheng, J., Sederberg, T. W., 2001. The mu-basis of a rational ruled surface. Comput. Aided Geom. Design 18 (1), 61–72] and showed that its resultant is the implicit equation of the surface, if the parametrization is generically injective. We generalize this result to the case of an arbitrary parametrization of a rational ruled surface. We also give a new proof for the corresponding theorem in the curve case and treat the reparametrization problem for curves and ruled surfaces. In particular, we propose a partial solution to the problem of computing a proper reparametrization for a rational ruled surface.