Plane Isometries As Complex Functions (original) (raw)
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 2times22\times 22times2 matrix. In the homogeneous coordinates all four transformations in their generality are represented by 3times33 \times 33times3 matrices. Complex variables supply an alternative representation for all four isometries [Erickson, 4.7].
Theorem
Every isometry of the Euclidean plane mathbbR2\mathbb{R}^2mathbbR2 viewed as the complex line mathbbC\mathbb{C}mathbbC is of one of the forms
(1) f(z)=alphaz+betaf(z)=\alpha z+\betaf(z)=alphaz+beta or f(z)=alphaoverlinez+betaf(z)=\alpha \overline{z}+\betaf(z)=alphaoverlinez+beta,
where alpha,betainmathbbC\alpha,\beta\in \mathbb{C}alpha,betainmathbbC, ∣alpha∣=1|\alpha|=1∣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+betaf(z) = z + \betaf(z)=z+beta is a translation by (vector) beta\betabeta, while f(z)=alphazf(z)=\alpha zf(z)=alphaz, with ∣alpha∣=1|\alpha|=1∣alpha∣=1 is a rotation around the origin through the angle of $arg(\alpha )$. Finally, f(z)=overlinezf(z)=\overline{z}f(z)=overlinez is the reflection in the xxx-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−1xyy^{-1}xyy−1xy. The first example is a rotation with an arbitrary center.
Assume we want to rotate the plane through an angle arg(alpha)arg(\alpha )arg(alpha) around point gamma\gammagamma. Start with translating the plane by −gamma-\gamma−gamma to place gamma\gammagamma at the origin. Next, rotate the plane by multiplying by alpha\alphaalpha, and then translate back by gamma\gammagamma. The result is f(z)=alpha(z−gamma)+gammaf(z) = \alpha (z - \gamma) + \gammaf(z)=alpha(z−gamma)+gamma.
On the other hand, function f(z)=alphaz+betaf(z) = \alpha z + \betaf(z)=alphaz+beta, with ∣alpha∣=1|\alpha|=1∣alpha∣=1 and alphane1\alpha\ne 1alphane1 can be rewritten as f(z)=alpha(z−gamma)+gammaf(z) = \alpha (z - \gamma) + \gammaf(z)=alpha(z−gamma)+gamma, where gamma=fracbeta1−alpha\gamma = \frac{\beta}{1-\alpha}gamma=fracbeta1−alpha. (For alpha=1\alpha = 1alpha=1, f(z)=alphaz+betaf(z) = \alpha z + \betaf(z)=alphaz+beta is a translation.)
In a similar way, we obtain a reflection in a line through the origin, tomegat\omegatomega, tinmathbbRt\in\mathbb{R}tinmathbbR, ∣omega∣=1|\omega|=1∣omega∣=1. First rotate tomegat\omegatomega to coincide with the xxx-axis, then reflect and, finally, rotate back the xxx-axis into the line tomegat\omegatomega: f(z)=omegaoverlineomega−1z=omega2overlinezf(z) = \omega \overline{\omega ^{-1}z}=\omega^{2}\overline{z}f(z)=omegaoverlineomega−1z=omega2overlinez.
Reflection in a line tomega+betat\omega + \betatomega+beta, which is parallel to tomegat\omegatomega, 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., iomegai\omegaiomega. Thus, we first write the "normalized" equation of the straight line as Iomegaz+isomega=0I\omega z+is\omega=0Iomegaz+isomega=0, where sss is the signed distance from the origin to the line and I2=−1I^{2}=-1I2=−1, and then perform the conjugation: f(z)=omega2overline(z−isomega)+isomega=omega2z+2isomegaf(z) = \omega^{2} \overline{(z-is\omega)}+is\omega=\omega^{2}z+2is\omegaf(z)=omega2overline(z−isomega)+isomega=omega2z+2isomega,
because omegacdotoverlineomega=∣omega∣2=1\omega\cdot\overline{\omega}=|\omega|^{2}=1omegacdotoverlineomega=∣omega∣2=1.
Glide reflection in a line omegaz+beta=0\omega z + \beta=0omegaz+beta=0 is a combination of reflection in the line and a translation by vector uomegau\omegauomega, uinmathbbRu\in\mathbb{R}uinmathbbR, parallel to the line: f(z)=omega2overline(z+uomega)+2isomega=omega2overlinez+2isomega+uomegaf(z) = \omega^{2} \overline{(z+u\omega)}+2is\omega=\omega^{2} \overline{z}+2is\omega+u\omegaf(z)=omega2overline(z+uomega)+2isomega=omega2overlinez+2isomega+uomega.
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)=overlinealphaz−overlinealphabetaf(z)=\overline{\alpha}z-\overline{\alpha}\betaf(z)=overlinealphaz−overlinealphabeta and f(z)=overlinealphaoverlinez−overlinealphabetaf(z)=\overline{\alpha}\overline{z}-\overline{\alpha}\betaf(z)=overlinealphaoverlinez−overlinealphabeta being inverses of f(z)=alphaz+betaf(z)=\alpha z+\betaf(z)=alphaz+beta and f(z)=alphaoverlinez+betaf(z)=\alpha \overline{z}+\betaf(z)=alphaoverlinez+beta, respectively. The group is not commutative. The orientation-preserving transformations f(z)=alphaz+betaf(z)=\alpha z+\betaf(z)=alphaz+beta form a subgroup; in which rotations f(z)=alphazf(z)=\alpha zf(z)=alphaz around the origin and translations f(z)=z+betaf(z)=z+\betaf(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)=alphaz+betaf(z)=\alpha z + \betaf(z)=alphaz+beta and f(z)=gammaz+deltaf(z)=\gamma z+\deltaf(z)=gammaz+delta is either a rotation or a translation (if alphagamma=1\alpha\gamma = 1alphagamma=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
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