Revisiting the <f>μ</f>-basis of a rational ruled surface

The <f>μ</f>-basis of a rational ruled surface <f>P(s,t)=P0(s)+tP1(s)</f> is defined in Chen et al. (Comput. Aided Geom. Design 18 (2001) 61) to consist of two polynomials <f>p(x,y,z,s)</f> and <f>q(x,y,z,s)</f> that are linear in <f>x</f>, <f>y</f>, <f>z</f>. It is shown there that the resultant of <f>p</f> and <f>q</f> with respect to <f>s</f> gives the implicit equation of the rational ruled surface; however, the parametric equation <f>P(s,t)</f> of the rational ruled surface cannot be recovered from <f>p</f> and <f>q</f>. Furthermore, the <f>μ</f>-basis thus defined for a rational ruled surface does not possess many nice properties that hold for the <f>μ</f>-basis of a rational planar curve (Comput. Aided Geom. Design 18 (1998) 803). In this paper, we introduce another polynomial <f>r(x,y,z,s,t)</f> that is linear in <f>x</f>, <f>y</f>, <f>z</f> and <f>t</f> such that <f>p</f>, <f>q</f>, <f>r</f> can be used to recover the parametric equation <f>P(s,t)</f> of the rational ruled surface; hence, we redefine the <f>μ</f>-basis to consist of the three polynomials <f>p</f>, <f>q</f>, <f>r</f>. We present an efficient algorithm for computing the newly-defined <f>μ</f>-basis, and derive some of its properties. In particular, we show that the new <f>μ</f>-basis serves as a basis for both the moving plane module and the moving plane ideal corresponding to the rational ruled surface. [Copyright &y& Elsevier]