Implicitizing rational surfaces of revolution using μ-bases

We provide a new technique for implicitizing rational surfaces of revolution using μ-bases. A degree n rational plane curve rotating around an axis generates a degree 2n rational surface. From a μ-basis p,q of this directrix curve, where μ=deg(p)≤deg(q)=n-μ, and a rational parametrization of the circle r(s)=(2s,1-s2,1+s2), we can easily generate three moving planes p*,q*,r* with generic bidegrees (1,μ),(1,n-μ),(2,0) that form a μ-basis for the corresponding surface of revolution. We show that this μ-basis is a powerful bridge connecting the parametric representation and the implicit representation of the surface of revolution. To implicitize the surface, we construct a 3n×3n Sylvester style sparse resultant matrix Rs,t for the three bidegree polynomials p*,q*,r*. Applying Gaussian elimination, we derive a 2n×2n sparse matrix Ss,t, and we prove that det(Ss,t)=0 is the implicit equation of the surface of revolution. Using Bezoutians, we also construct a 2(n-μ)×2(n-μ) matrix Bs,t, and we show that det(Bs,t)=0 is also the implicit equation of the surface of revolution. Examples are presented to illustrate our methods.