The present paper investigates two-parameter families of spheres in R3 and their corresponding two-dimensional surfaces in R4. Considering a rational surface in R4, the envelope surface of the corresponding family of spheres in R3 is typically non-rational. Using a classical spheregeometric approach we prove that the envelope surface and its offset surfaces admit rational parameterizations if and only if is a rational sub-variety of a rational isotropic hyper-surface in R4. The close relation between the envelope surfaces and rational offset surfaces in R3 is elaborated in detail. This connection leads to explicit rational parameterizations for all rational surfaces in R4 whose corresponding two-parameter families of spheres possess envelope surfaces admitting rational parameterizations. Finally we discuss several classes of surfaces sharing this property.
author = "Martin Peternell",
title = "Rational Two-Parameter Families of Spheres and Rational
journal = "J. Symbolic Computation",
year = 2010,
volume = 45,
pages = "1-18",