CHAPTER II - Orthogonal Transformations

Abstract

In geometry, as opposed to kinematics, a displacement is not regarded as an actual process of motion from one point to another but merely as a correspondence between the points occupied by the figure in its initial and final positions; such an approach allows viewing displacements in geometry as mappings that take intervals into equal intervals. From the geometric point of view, such mappings are the simplest because they preserve both the dimensions and the shapes of figures and change only their position. These transformations are called orthogonal mappings or orthogonal transformation. This chapter provides an overview of such transformations. An orthogonal mapping of a plane onto itself is called an orthogonal transformation of the plane. The product of any two orthogonal transformations is itself an orthogonal transformation and the same holds for the inverse of an orthogonal transformation. The set of all orthogonal transformations of the plane forms a group called the orthogonal group of the plane. In a similar way, the orthogonal transformations of space may be defined and shown that they form a group. The chapter discusses some properties of orthogonal mappings and discusses the important geometric concept of orientation. It describes the orthogonal transformations of the first and second kinds. The chapter also discusses the fundamental types of orthogonal transformation, in terms of which every such transformation can be expressed. It describes three special types of orthogonal transformation: translation, reflection, and rotation and discusses that any plane orthogonal transformation may be represented as a product of such special transformations.