On the Affine Image of a Rational Surface of Revolution

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.

Intersections of Surfaces of Revolution

“…Finally we consider the intersection IS of general surfaces of revolution RS (1) and RS (2) given by the parametric representations…”

“…x (1) (u i ) = r 1 (u 1 ) cos u 2 e 1 L,1 + r 1 (u 1 ) sin u 2 e 2 L,1 + h 1 (u 1 ) e 3 L,1 and x (2) (v i ) = r 2 (v 1 ) cos v 2 e 1 L,2 + r 2 (v 1 ) sin v 2 e 2 L,2 + h 2 (v 1 ) e 3 L,2 . Since the functions r 1 , r 2 , h 1 and h 2 satisfy the conditions in (2.17), at each u i or v i at least one of them has a local inverse.…”

“…Nevertheless, they are still interesting for research. Some papers that explore their properties are [1,2,3,6]. Some of the papers study the intersections of surfaces of revolution are [8,9,21].…”