On the Affine Image of a Rational Surface of Revolution

Juan G. Alcázar
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Juan G. Alcázar: Departamento de Física y Matemáticas, Universidad de Alcalá, E-28871 Madrid, Spain

Mathematics, 2020, vol. 8, issue 11, 1-17

Abstract: We study the properties of the image of a rational surface of revolution under a nonsingular affine mapping. We prove that this image has a notable property, namely that all the affine normal lines , a concept that appears in the context of affine differential geometry , created by Blaschke in the first decades of the 20th century, intersect a fixed line. Given a rational surface with this property, which can be algorithmically checked, we provide an algorithmic method to find a surface of revolution, if it exists, whose image under an affine mapping is the given surface; the algorithm also finds the affine transformation mapping one surface onto the other. Finally, we also prove that the only rational affine surfaces of rotation , a generalization of surfaces of revolution that arises in the context of affine differential geometry, and which includes surfaces of revolution as a subtype, affinely transforming into a surface of revolution are the surfaces of revolution, and that in that case the affine mapping must be a similarity.

Keywords: surface of revolution; affine differential geometry; affine equivalence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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