On the Affine Image of a Rational Surface of Revolution

Mathematics

JuanG. Alcázar¹ 

[1] Departamento de Física y Matemáticas, Universidad de Alcalá, E-28871 Madrid, Spain;

关键词: surface of revolution;    affine differential geometry;    affine equivalence;   

DOI  :  10.3390/math8112061

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown