Classification of Plane Isometries

Abstract

We have classified all the even isometries as translations or rotations. An odd isometry is a reflection or a product of three reflections. Only those odd isometries σ_c σ_b σ_a where a, b, c are neither concurrent nor have a common perpendicular remain to be considered. Although it seems there might be many cases, depending on which of a, b, c intersect or are parallel to which, we shall see this turns out not to be the case. However, we begin with the special case where a and b are perpendicular to c. Then σ_b σ_a is a translation or glide and σ_c is, of course, a reflection. If a and b are distinct lines perpendicular to line c, then σ_c σ_b σ_a is called a glide reflection with axis c. We might as well call line m the axis of σ_m as the reflection and the glide reflection then share the property that the midpoint of any point P and its image under the isometry lies on the axis. To show this holds for the glide reflection, suppose P is any point. See Figure 8.1. Let line l be the perpendicular from P to c. Then there is a line m perpendicular to c such that σ_b σ_a = σ_m σ_l. If M is the intersection of m and c, then P and M are distinct points such that σ_cσ_bσ_a(P)=σ_cσ_mσ_l(P)=σ_cσ_m(P)= σ_M(P)≠P.