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

.