An Algorithm to Compute the mu-Bases of Rational Parametric Surfaces with Respect to One Variable

The method of moving surfaces is an effective tool to implicitize rational parametric surfaces, and it has been extensively studied in the past two decades. An essential step in surface implicitization using the method of moving surfaces is to compute a mu-basis of a parametric surface with respect to one variable. The mu-basis is a minimal basis of the syzygy module of a univariate polynomial matrix with special structure defined by the parametric equation of the rational surface. In this paper, we present an efficient algorithm to compute the mu-basis of a parametric surface with respect to a variable based on the special structure of the corresponding univariate polynomial matrix. Analysis on the computational complexity of the algorithm is also provided. Experiments demonstrate that our algorithm is much faster than the general method to compute the mu-bases of arbitrary polynomial matrices and outperforms the F-5 algorithm based on Grabner basis computation for relatively low degree rational surfaces.