On the Theory of Surfaces in the Affine Space: II. Generalized Affine Moulding Surfaces and Affine Surfaces of Revolution

Abstract

A Characteristic Property of Affine Surfaces of Revolution. 17. In my former paper (1) I investigated a class of surfaces, called "affine moulding surfaces" , having a system of curves on parallel planes which are at the same time the curves of contact of tangent planes drawn from any point on a curve. The curves on parallel planes are defined as "parallel curves" of the surface while the con jugate system of the "parallel curves" is called "meridian curve". A special class of the affine moulding surface, the affine surface of revolu tion, is defined by the fact that the affine surface-normal falls in the osculating plane of the meridian curve. This class is, however, identical with that obtained by Dr. W. Suss (2) from another standpoint. This suggests us to prove the identity of these two definitions of Dr. Suss and of mine. First we prove Theorem 18. If the affine surface-normals of a surface all intersect a given (proper) straight line and if the meridians are lines of shadow, then the surface must necessarily be an affine surface of revolution.