Parameterization of algebraic space curves

Abstract

This paper describes algorithms using Groebner basis techniques for parameterizing algebraic space curves. It is shown that the Groebner basis of the ideal of a non-singular rational space curve of degree n in general position, taken with respect to an appropriate elimination order, has n + 1 elements whose degrees and structure can be precisely described. The space curve can be parameterized by taking a birational projection to the plane and parameterizing the resulting plane curve. The Groebner basis just described contains all the information necessary to deal with the singularities introduced by projection, so that explicit calculations relating to these singularities can be avoided. The modifications necessary to take care of the cases of a singular space curve, or a curve not in general position, are given. An appendix sketches a proof of the result that the projection of a non-singular rational, non-planar space curve from any point is birational.