Rational two-parameter families of spheres and rational offset surfaces

• Autores: Martin Peternell
• Localización: Journal of symbolic computation, ISSN 0747-7171, Vol. 45, Nº 1, 2010, págs. 1-18
• Idioma: inglés
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• Resumen

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