CHAPTER IV - Affine Transformations

Abstract

The group of similarity transformations of the plane is a subgroup of a group of more general transformations that preserve collinearity and parallelism but not, in general, the lengths of segments and the sizes of angles or areas. These transformations are called affine transformations. This chapter presents an overview of affine transformations. This chapter discusses the affine mappings and transformations of the plane. Under an affine mapping, any three noncollinear points have noncollinear images. The chapter also presents a proof for this theorem. It provides some examples of affine transformations and mappings of a plane. Under similarity transformations, angles are preserved but not lengths, unless the transformation is orthogonal. The simplest example of an affine transformation in which both lengths and angles change is provided by skew reflection. The chapter reviews some properties of affine mappings through theorems and discusses the representation of any affine transformation as a product of affine transformations of the simplest types. Any affine transformation of the plane can be represented as the product of a similarity transformation and an affinity. An affinity is any affine transformation leaving some line point-wise invariant. The chapter discusses the noninvariance of lengths of segments under affine mappings and the application of affine transformations to the investigation of properties of the ellipse. It also discusses the affine transformations in coordinates, the affine classification of quadratic curves, and the affine transformations of space.