Chapter 2 - Projective and Affine Geometry over Division Rings

Abstract

This chapter discusses projective and affine geometry over division rings. Projective and affine geometry is the geometry of subspaces (or cosets of subspaces) of vector spaces over division rings. The chapter constructs the projective and affine spaces over division rings synthetically, and gives the connection between them, with some historical comments. A description of the axiomatic approach is provided and the Veblen–Young axioms which characterize projective spaces of dimension at least 3. The chapter defines projective planes, and describes the connection between Desargues' Theorem and coordinatization, which is important in the general coordinatization theorem. It discusses Pappus' Theorem, which implies Desargues' Theorem and is equivalent to commutativity of the coordinatizing division ring. It explains how the axioms have to be modified to describe affine spaces. It provides the description of collineations and correlations (dualities) of projective and affine spaces, including the structure of the collineation groups and the classification of polarities.