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 $σ$c$σ$b$σ$a(P)=$σ$c$σ$m$σ$l(P)=$σ$c$σ$m(P)= $σ$M(P)≠P.