If we limit the set of isometric transformations to just translations and rotations, then we may call the resulting set the rigid-body motions, or simply motions. Remember, in the theory of transformations there is an initial or original position and an image or final position. There are no intermediate positions. There is no path. However, with the motion transformations we now relax that outlook. We begin this chapter with the translation transformation and consider the Cartesian translation equations, vector-defined translations, a succession of translations, unequal translations over two points, invariants, and finally the translation group. We study the rotation transformation in two and three dimensions, equivalent rotations (where we require the services of eigenvalues and eigenvectors), products of rotations, and the rotation group. Next, we consider composite motion, a combination of translation and rotation transformations. Here we address the problem of how to mathematically express a composite motion in a computationally efficient and convenient way. Homogeneous coordinates and the homogeneous transformation matrix offer a solution. In the final section of this chapter, we briefly explore kinematic transformations. All of these transformations operate within the standard, right-hand Cartesian coordinate system, unless specified otherwise.

4.1 Translation

Vectors and translation, translation of a line, coordinate system translation, a succession of translations, unequal translations over two points, invariants under translation, the translation group

A translation is a mapping given by Cartesian equations of the form

(4.1)

The translation transformation of a point within the framework of a coordinate system is not very exciting...