Abstract
The present paper investigates two-parameter families of spheres in R 3 and their corresponding two-dimensional surfaces Φ in R 4 . Considering a rational surface Φ in R 4 , the envelope surface Ψ of the corresponding family of spheres in R 3 is typically non-rational. Using a classical sphere-geometric approach, we prove that the envelope surface Ψ and its offset surfaces admit rational parameterizations if and only if Φ is a rational sub-variety of a rational isotropic hyper-surface in R 4 . The close relation between the envelope surfaces Ψ and rational offset surfaces in R 3 is elaborated in detail. This connection leads to explicit rational parameterizations for all rational surfaces Φ in R 4 whose corresponding two-parameter families of spheres possess envelope surfaces admitting rational parameterizations. Finally we discuss several classes of surfaces sharing this property.
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