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|=1alpha=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|=1alpha=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}xyy1xy. 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-\gammagamma 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(zgamma)+gamma.

On the other hand, function f(z)=alphaz+betaf(z) = \alpha z + \betaf(z)=alphaz+beta, with ∣alpha∣=1|\alpha|=1alpha=1 and alphane1\alpha\ne 1alphane1 can be rewritten as f(z)=alpha(z−gamma)+gammaf(z) = \alpha (z - \gamma) + \gammaf(z)=alpha(zgamma)+gamma, where gamma=fracbeta1−alpha\gamma = \frac{\beta}{1-\alpha}gamma=fracbeta1alpha. (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|=1omega=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)=omegaoverlineomega1z=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(zisomega)+isomega=omega2z+2isomega,

because omegacdotoverlineomega=∣omega∣2=1\omega\cdot\overline{\omega}=|\omega|^{2}=1omegacdotoverlineomega=omega2=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)=overlinealphazoverlinealphabeta and f(z)=overlinealphaoverlinez−overlinealphabetaf(z)=\overline{\alpha}\overline{z}-\overline{\alpha}\betaf(z)=overlinealphaoverlinezoverlinealphabeta 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

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

Plane Isometries As Complex Functions

  1. Plane Isometries
  2. Reflection
  3. Translation
  4. Rotation
  5. Glide Reflection

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