Implicitizing rational surfaces of revolution using mu-bases
We provide a new technique for implicitizing rational surfaces of revolution using mu-bases. A degree n rational plane curve rotating around an axis generates a degree 2n rational surface. From a mu-basis p,q of this directrix curve, where mu = deg(p) <= deg(q) = n - mu, and a rational parametrization of the circle r(s) = (2s, 1 - s(2), 1 + s(2)), we can easily generate three moving planes p*, q*, r* with generic bidegrees (1, mu), (1,n - mu), (2, 0) that form a mu-basis for the corresponding surface of revolution. We show that this mu-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 x 3n Sylvester style sparse resultant matrix R-s,R-t for the three bidegree polynomials p*,q*,r*. Applying Gaussian elimination, we derive a 2n x 2n sparse matrix S-s,S-t, and we prove that det(S-s,S-t) = 0 is the implicit equation of the surface of revolution. Using Bezoutians, we also construct a 2(n - mu) x 2(n - mu) matrix B-s,B-t and we show that det(B-s,B-t) = 0 is also the implicit equation of the surface of revolution. Examples are presented to illustrate our methods. (C) 2012 Elsevier B.V. All rights reserved.
implicitization surface of revolution mu-basis resultant matrix