Lecture 5 37
x = Ix
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
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: