New Foundations of Projective and Affine Geometry

Abstract

Projective geometry is often called geometry of projection and section (Geometrie des Verbindens und Schneidens). In this paper foundations of projective geometry are given in terms of these two operations. We start from a single class of undefined entities, corresponding to the linear parts of a space, and two undefined operations denoted by + and ., corresponding to the join and the intersection, respectively, of these linear parts. Thus if A, B are two undefined entities, A + B corresponds to the least dimensional part of which both A and B are parts, while A • B corresponds to the highest dimensional part which is part both of A and of B. In this way we obtain a far-reaching analogy with abstract algebra where, in defining a field, one also starts with a Single class of undefined elements and two undefined operations, addition and multiplication. Moreover, we obtain an analogy with the algebra of logics, in particular with the calculus of classes.1 In fact, this paper presents what might be called an algebra of elementary geometry.