Revisiting the μ-basis of a rational ruled surface

Abstract

The μ-basis of a rational ruled surface $\text{P}\text{(s,t)=}\text{P}{}_0\text{(s)+t}\text{P}{}_1\text{(s)}$ is defined in Chen et al. (Comput. Aided Geom. Design 18 (2001) 61) to consist of two polynomials p(x,y,z,s) and q(x,y,z,s) that are linear in x, y, z. It is shown there that the resultant of p and q with respect to s gives the implicit equation of the rational ruled surface; however, the parametric equation $\text{P}\text{(s,t)}$ of the rational ruled surface cannot be recovered from p and q. Furthermore, the μ-basis thus defined for a rational ruled surface does not possess many nice properties that hold for the μ-basis of a rational planar curve (Comput. Aided Geom. Design 18 (1998) 803). In this paper, we introduce another polynomial r(x,y,z,s,t) that is linear in x, y, z and t such that p, q, r can be used to recover the parametric equation $\text{P}\text{(s,t)}$ of the rational ruled surface; hence, we redefine the μ-basis to consist of the three polynomials p, q, r. We present an efficient algorithm for computing the newly-defined μ-basis, and derive some of its properties. In particular, we show that the new μ-basis serves as a basis for both the moving plane module and the moving plane ideal corresponding to the rational ruled surface.