Affine differential geometry and affine rotation surfaces: algebraic surfaces invariant under non-Euclidean affine rotations

Abstract

Affine rotation surfaces are a generalization of the well-known surfaces of revolution. Affine rotation surfaces arise naturally within the framework of affine differential geometry, a field started by Blaschke in the first decades of the past century. Affine rotations are the affine equivalents of Euclidean rotations, and include certain shears as well as Euclidean rotations. Affine rotation surfaces are surfaces invariant under affine rotations. In this paper, we analyze several properties of algebraic affine rotation surfaces and, by using some notions and results from affine differential geometry, we develop an algorithm for determining whether or not an algebraic surface given in implicit form, or in some cases in rational parametric form, is an affine rotation surface. We also show how to find the axis of an affine rotation surface. Additionally, we discuss several properties of affine spheres, analogues of Euclidean spheres in the context of affine differential geometry.