Computing μ-bases of rational curves and surfaces using polynomial matrix factorization

The μ-bases of rational curves/surfaces are newly developed tools which play an important role in connecting parametric forms and implicit forms of the rational curves/surfaces. They provide efficient algorithms to implicitize rational curves/surfaces as well as algorithms to compute singular points of rational curves and to reparametrize rational ruled surfaces. In this paper, we present an efficient algorithm to compute the μbasis of a rational curve/surface by using polynomial matrix factorization followed by a technique similar to Gaussian elimination. The algorithm is shown superior than previous algorithms to compute the μ-basis of a rational curve, and it is the only known algorithm that can rigorously compute the μ-basis of a general rational surface. We present some examples to illustrate the algorithm.