A contribution to the theory of fundamental transformations of surfaces

Abstract

Two surfaces are said to be related to one another by a fundamental transformation, that is, by a transformation F, if the developables of the congruence of lines joining corresponding points on the surfaces cut the surfaces in conjugate nets of curves. It is assumed that neither of these nets is a focal net of the congruence. The nets on the surfaces are also said to correspond by the transformation F. Although many well known transformations of surfaces are special types of transformations F, the general case was treated in detail but recently, by Eisenhartt and Jonas.t In a recent paper Graustein? introduced into the study of these transformations a projective invariant which was the generalization of the invariant of a parallel map.11 Certain important theorems concerning this invariant were obtained whose nature indicates that transformations F can be investigated to advantage by means of it. We call this invariant the invariant C. When studied in terms of tangential coordinates, transformations F present a complete duality among the elements involved. In this way, a second invariant, the invariant H, is obtained which is dual to the invariant C. The invariant C is equal to the cross ratio in which a pair of corresponding points of the surfaces in the relation F is divided by the focal points of the line joining them. Dually, the invariant H is equal to the cross ratio in which a pair of corresponding tangent planes to the two surfaces is divided by the focal planes through their line of intersection.