Chapter 3 - metric affine spaces

Abstract

This chapter focuses on metric affine spaces. A metric affine space is an affine space (X, V, k) where V is a metric vector space. Two affine subspaces of X are called orthogonal—perpendicular—if their direction spaces are orthogonal. The chapter discusses a theorem that considers the rigid motions which leave a point of X fixed. It states that these motions form a group and describe the group fully. It follows immediately from the theorem that the structure of the group of motions which leave a point fixed does not depend on the choice of that point. In Euclidean geometry, it is customary to define a Euclidean transformation as a one-to-one function of X onto itself that preserves distance. In other words, one never mentions that a Euclidean transformation has to be an affine transformation. Each theorem for metric vector spaces has an analogue for metric affine spaces. Usually, all that one has to do to find the analogue is to investigate the behavior of the translations of X. The Cartan-Dieudonné theorem for affine spaces is also proven in the chapter.