Computing the shapes arising in a family of space rational curves depending on one parameter

In a previous work (Alcazar, 2009) we addressed the problem of determining the topology types arising in a family of plane rational curves depending on one parameter, defined by means of a rational parametrization. In this paper, starting from the ideas in Alcazar (2009) we address the analogous problem for families of space rational curves, also defined by means of a rational parametrization. Hence, the main result in the paper is an algorithm for computing the (finitely many) real values of the parameter where the topology of the family may change. The algorithm has been implemented in the computer algebra system Maple 13, with good practical results; several examples and timings are provided.

► We compute the shapes in a family of rational curves depending on a parameter.
► The algorithm relies in a “good position” for the family.
► Such a good position is achievable by applying a generic change of coordinates.
► The algorithm is tested over a number of examples.
► Timings, details are provided.