Rationality Analysis and Direct Parametrization of Generalized Offsets to Quadrics

Abstract.

In this paper, working over algebraically closed fields of characteristic zero, we present formal proofs for a complete analysis of the rationality of generalized offsets to all irreducible quadrics, and we show how to derive rational parametrizations of the offsets, when they are rational, from the parametrization of the original quadric. This is achieved by means of a surface, constructed without implicitizating, that is either linear in one variable or tubular.

More precisely, we prove that generalized offsets to elliptic and hyperbolic paraboloids are rational, generalized offsets to parabolic cylinders are rational, and generalized offsets to hyperbolic cylinders are irreducible and not rational. Also we prove that generalized offsets to elliptic cylinders of revolution have two rational components, and that generalized offsets to non-revolution elliptic cylinders are irreducible and not rational. Furthermore, we state that generalized offsets to cones of revolution have two rational components and to non-revolution cones are irreducible and not rational. In addition, an offset to the sphere has two rational components and to an ellipsoid is rational. Finally, we prove that generalized offsets to one and two-sheeted hyperboloids are rational.