Overview

Reflections and the more complex symmetry transformations differ from rigid body motions only because parity or handedness may be reversed, as in a mirror reflection. However, the size and shape of an object or figure do not change under these transformations, except in special cases such as oblique reflections. In fact, we saw in Chapter 2 that an appropriate set of reflections will generate any rigid motion.

There are many reasons for studying the reflection transformations. For example, only by supplementing the direct isometries (rigid motions) with reflections can we generate all the crystal structures that we find in nature. Reflection transformations also play a significant role in our study of symmetry, where centers of symmetry and lines of symmetry are a natural result. Consequently, the mathematics of reflections has considerable aesthetic appeal. As we shall see, reflections are orientation reversing; therefore, we must exclude them from the family of allowable topological deformations. The distinguishing characteristic of a reflection is that the determinant of its transformation matrix, R _f, is equal to minus one:

(5.1)

and this is what tells us that every reflection is an orientation reversing one.

Reflection transformations fixing a plane in space produce what we call a mirror image. Reflection in the plane and in space can also fix a point or a line. If it fixes a point, it is a central inversion or, more simply, an inversion. We began our study of reflections in Chapter 2, using synthetic definitions and Cartesian equations.