Lecture 5 37 x = Ix y z Iy Iz x y z Ix Iy Iz Reflection—a line of fixed points Glide reflection—no fixed points Figure 1.23. Orientation reversing isometries. approach as in case 1, the reflection through takes x to Ix and y to Iy since it reverses orientation, I is exactly this reflection. It takes two parameters to specify a line, and hence a reflection, so the space of reflections is two-dimensional. Case 4 : An orientation reversing isometry with no fixed point is a glide reflection. Let T be the unique translation that takes x to Ix. Then I = R◦T where R = I ◦T−1 is an orientation reversing isometry which fixes Ix. By the above, R must be a reflection through some line . Decompose T as T1 T2, where T1 is a translation by a vector perpendicular to , and T2 is a translation by a vector parallel to . Then I = R T1 T2, and R T1 is a reflection through a line parallel to , hence I is the composition of a translation T2 and a reflection R T1 which commute that is, a glide reflection. A glide reflection is specified by three parameters hence the space of glide reflections is three-dimensional, so almost every orientation reversing isometry is a glide reflection, and hence has no fixed point. The group Isom(R2) is a topological group with two components one component comprises the orientation preserving isometries, the other the orientation reversing isometries. From the above discussions of how many parameters are needed to specify an isometry, we see that the group is three-dimensional in fact, it has a nice embedding into the group GL(3, R) of invertible 3 × 3 matrices: Isom(R2) = O(2) R2 0 1 : R2 1 R2 1 .
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