# Plane Isometries As Complex Functions

There are four classes of plane isometries: translation, reflection, rotation, and glide reflection. Rotation around the origin, or reflections in the coordinate axes could be compactly represented by a $2\times 2$ matrix. In the homogeneous coordinates all four transformations in their generality are represented by $3 \times 3$ matrices. Complex variables supply an alternative representation for all four isometries [Erickson, 4.7].

### Theorem

Every isometry of the Euclidean plane $\mathbb{R}^2$ viewed as the complex line $\mathbb{C}$ is of one of the forms

(1)

$f(z)=\alpha z+\beta$ or $f(z)=\alpha \overline{z}+\beta$,

where $\alpha,\beta\in \mathbb{C}$, $|\alpha|=1$. The first function is orientation-preserving; the second is orientation-reversing.

For a proof, observe that a few specific cases of (1) have an immediate geometric interpretation due to the properties of operations over complex numbers. For example, $f(z) = z + \beta$ is a translation by (vector) $\beta$, while $f(z)=\alpha z$, with $|\alpha|=1$ is a rotation around the origin through the angle of $arg(\alpha )$. Finally, $f(z)=\overline{z}$ is the reflection in the $x$-axis.

Ultimately, the set of all the plane isometries is a group under the product of functions - in this case, the composition, i.e., a successive execution of two operations (1). With this in mind, I shall repeatedly invoke group conjugation, $y^{-1}xy$. The first example is a rotation with an arbitrary center.

Assume we want to rotate the plane through an angle $arg(\alpha )$ around point $\gamma$. Start with translating the plane by $-\gamma$ to place $\gamma$ at the origin. Next, rotate the plane by multiplying by $\alpha$, and then translate back by $\gamma$. The result is

$f(z) = \alpha (z - \gamma) + \gamma$.

On the other hand, function $f(z) = \alpha z + \beta$, with $|\alpha|=1$ and $\alpha\ne 1$ can be rewritten as $f(z) = \alpha (z - \gamma) + \gamma$, where $\gamma = \frac{\beta}{1-\alpha}$. (For $\alpha = 1$, $f(z) = \alpha z + \beta$ is a translation.)

In a similar way, we obtain a reflection in a line through the origin, $t\omega$, $t\in\mathbb{R}$, $|\omega|=1$. First rotate $t\omega$ to coincide with the $x$-axis, then reflect and, finally, rotate back the $x$-axis into the line $t\omega$:

$f(z) = \omega \overline{\omega ^{-1}z}=\omega^{2}\overline{z}$.

Reflection in a line $t\omega + \beta$, which is parallel to $t\omega$, is by first translating the line to pass through the origin, taking a reflection and then translating the plane back. A caveat, though, is that the translation has to be in the direction perpendicular to the line, i.e., $i\omega$. Thus, we first write the "normalized" equation of the straight line as $I\omega z+is\omega=0$, where $s$ is the signed distance from the origin to the line and $I^{2}=-1$, and then perform the conjugation:

$f(z) = \omega^{2} \overline{(z-is\omega)}+is\omega=\omega^{2}z+2is\omega$,

because $\omega\cdot\overline{\omega}=|\omega|^{2}=1$.

Glide reflection in a line $\omega z + \beta=0$ is a combination of reflection in the line and a translation by vector $u\omega$, $u\in\mathbb{R}$, parallel to the line:

$f(z) = \omega^{2} \overline{(z+u\omega)}+2is\omega=\omega^{2} \overline{z}+2is\omega+u\omega$.

Note here that the equation confirms the notion that the order in which a point is reflected in a line and translated in the direction parallel to the line is not important: the two operations commute.

It is clear that transformations (1) form a group, with $f(z)=\overline{\alpha}z-\overline{\alpha}\beta$ and $f(z)=\overline{\alpha}\overline{z}-\overline{\alpha}\beta$ being inverses of $f(z)=\alpha z+\beta$ and $f(z)=\alpha \overline{z}+\beta$, respectively. The group is not commutative. The orientation-preserving transformations $f(z)=\alpha z+\beta$ form a subgroup; in which rotations $f(z)=\alpha z$ around the origin and translations $f(z)=z+\beta$ are subgroups in their own right. In turn, the group of all isometries is a subgroup of the group of the Möbius transforms. In general, the product of two rotations $f(z)=\alpha z + \beta$ and $f(z)=\gamma z+\delta$ is either a rotation or a translation (if $\alpha\gamma = 1$). It could be verified that translations and rotations could be affected by a pair of reflections; for translation, the axes of the reflections are parallel; for rotations, they cross. It follows that glide reflections could be obtained as the product of three reflections.

### References

1. M. Erickson, Beautiful Mathematics, MAA, 2011