Geometrical Representations

Abstract

This chapter discusses geometrical representations. It presents an overview of linear representations. These include geometries with linear but no metric structure, such as projective and affine geometry, and ones with compatible linear and metric structures, such as Hilbert geometry and Minkowski geometry. The chapter also focuses on groups of permissible transformations. It discusses metric spaces that fall outside the main lines of development of geometry but that have been proposed or used as representing structures for proximity relations. The pure theory of projective geometry is one of the greatest jewels of human intellect. The important geometrical concepts that can be defined in affine geometry are the spans and intersections of linear varieties.