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

M. Erickson, Beautiful Mathematics, MAA, 2011

Plane Isometries As Complex Functions

72520424