Implicitizing rational surfaces of revolution using μ-bases Original Research Article

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 image of this directrix curve, where image, and a rational parametrization of the circle image, we can easily generate three moving planes image with generic bidegrees image 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 image Sylvester style sparse resultant matrix image for the three bidegree polynomials image. Applying Gaussian elimination, we derive a image sparse matrix image, and we prove that image is the implicit equation of the surface of revolution. Using Bezoutians, we also construct a image matrix image, and we show that image is also the implicit equation of the surface of revolution. Examples are presented to illustrate our methods.