Chapter II - Transformation Geometry

Abstract

This chapter presents the introduction and history of transformation geometry. The symmetries of a figure form a group. This means that the combined effect of two symmetries is symmetry, the inverse of symmetry is symmetry, and the identity map is symmetry. Symmetry groups can be generalized to transformation groups. A nonempty set of maps from a space to itself is called a transformation group if it is closed under composition, and taking inverses. Geometry is the study of the properties of a space that are preserved by a transformation group. Euclidean geometry is the study of those properties such as length, and angle that are preserved by isometries. Different transformation groups preserve different properties and determine different geometries. This point of view, called the Erlanger Program, extended the idea of invariants from algebra to geometry and provided a theoretical framework for the special theory of relativity.