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,qp,q of this directrix curve, where μ=deg(p)⩽deg(q)=n−μμ=deg(p)⩽deg(q)=n−μ, and a rational parametrization of the circle r(s)=(2s,1−s2,1+s2)r(s)=(2s,1−s2,1+s2), we can easily generate three moving planes p⁎,q⁎,r⁎p⁎,q⁎,r⁎ with generic bidegrees (1,μ),(1,n−μ),(2,0)(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×3n3n×3n Sylvester style sparse resultant matrix Rs,tRs,t for the three bidegree polynomials p⁎,q⁎,r⁎p⁎,q⁎,r⁎. Applying Gaussian elimination, we derive a 2n×2n2n×2n sparse matrix Ss,tSs,t, and we prove that det(Ss,t)=0det(Ss,t)=0 is the implicit equation of the surface of revolution. Using Bezoutians, we also construct a 2(n−μ)×2(n−μ)2(n−μ)×2(n−μ) matrix Bs,tBs,t, and we show that det(Bs,t)=0det(Bs,t)=0 is also the implicit equation of the surface of revolution. Examples are presented to illustrate our methods.

► We compute a μ-basis for a surface of revolution from a μ  -basis for its directrix.
► We construct sparse resultant matrices for three bivariate polynomials of bidegrees (1,μ),(1,n−μ),(2,0)(1,μ),(1,n−μ),(2,0).
► We provide compact expressions for the implicit equation of a surface of revolution.