Rational two-parameter families of spheres and rational offset surfaces |
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Authors: | Martin Peternell |
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Affiliation: | Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstrasse 8-10, Vienna, Austria |
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Abstract: | The present paper investigates two-parameter families of spheres in R3 and their corresponding two-dimensional surfaces Φ in R4. Considering a rational surface Φ in R4, the envelope surface Ψ of the corresponding family of spheres in R3 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 R4. The close relation between the envelope surfaces Ψ and rational offset surfaces in R3 is elaborated in detail. This connection leads to explicit rational parameterizations for all rational surfaces Φ 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. |
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Keywords: | Space of spheres Envelope surface Minkowski space Rational offset surface |
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