Last edited 3apr15. Revised with thanks to Tony Winkler, 28jul11, Zheyuan Fan, 29jul11, and Saman Moniri, 31jul13.

Lesson on Hyperbolic Translations

$\C$ 2010, 2015 Prof. George K. Francis, Mathematics Department, University of Illinois. \begin{document} \maketitle \section{Introduction} Of all the rigid motions, \texit{translation} surely is the most fundamental. At the earliest age, a child is applauded when she moves one object from here to there. Rotations and reflections come much later. Euclid recognized this by his second Postulate, namely that a given line segment may be \textit{prolonged} (in both directions) indefinitely into an \text{infinite line}. The Greeks had an aversion of using the notion of infinity, in part because it was controversial. Even today, poorly instructed school children misinterpret that word infinity as the name of a very large number, as in million, billion, trillion, gazillion, infinity. No wonder they feel embarrassed when the teacher forbids them to ask why $\infity +1 $ isn't any bigger. If for no other reason, learning the calculus is a intelligent requirement of every educated person. However, the Greeks would have been quite happy with an alternative word for an infinite line, namely an \textit{unbounded line}. Recall that in the very first construction in this course, the one leading to a proof of the Absolute Exterior Angle Theorem, we used this concept. In this lesson we learn which of the hyperbolic transformations come closest to our intuitive understanding of translations. A \textit{hyperbolic translation} turn out to be the negative of an involution, which is the rotation of an involution about the origin by $180^o$. Hence every hyperbolic transformation can also be written as a rotated translation. Thus, the rule that every Euclidean motion is the composition of rotation with a translation (careful, the order matters) persists also in nonEuclidean geometry. We will compare Euclidean tranlation with hyperbolic translation, discovering many similarities, but also some significant differences. In particular, by looking at what I like to call the \textit{Saccheri Railroad} we will lay to rest once and for all the persistent misconception that to be parallel, lines should be equidistant from each other. Indeed, this property is entirely Euclidean. In the hyperbolic plane, equidistant curves cannot both be hyperbolic lines. \section{Hyperbolic Translations} \textbf{Denition of an hTranslation:} \[ T_\alpha = \frac{z+\alpha}{\bar{\alpha}z +1} , \, |\alpha|<1 .\] Observe the similarity of this definition with that of an involution in a prior lesson. Indeed
Question 1.
Show that \[ T_\alpha (z) = - V_{-\alpha}(z) .\]
Question 2.
Given $|\alpha| < 1$ and angle $\theta$, find $|\omega|<1$ and angle $\psi$ so that \[ e^{i\theta}V_\alpha(z) = e^{i\psi}T_\omega(z). \]
Question 3.
Show that \begin{eqnarray*} \mbox{ While }T_\alpha(0) &=& \alpha \\ \mbox{ and } T_\alpha(-\alpha) &=& 0 \\ \mbox{ BUT } T_\alpha(\alpha) &=& 2\alpha \frac{2\alpha}{1+|\alpha|^2} \\ \end{eqnarray*} To compute an explicit form for the inverse of a translation we have two choices of how to proceed. We can solve $ w= \frac{z+\alpha}{\bar{\alpha}z+1 $ directly for $z = \frac{w-\alpha}{-\bar{\alpha}w+1}$, or avoid the simple algebra and apply previously discovered properties of rotated involutions. The latter is a good review, but the former is quicker here. We can also calculate that the two fixed points of translation are the two points, $ \pm \alpha/|\alpha}$ where the diameter through $\alpha$ meets the unit circle. Thus $T_\alpha$ has no fixed \textit{hyperbolic points} Finally, one should really reserve the term "translation" for a transformation that is defined by its taking any point to some other point, not just $0 \mapsto \alpha$. \textit{Definition Translation from Here to There} \[ T_{\alpha, \omega} = T_\beta \circ T_\alpha^{-1} (z) = T_\beta \circ T_{-\alpha} (z) \] Compare this to the definition of a Euclidean translation in terms of vectors. So we begin to see the similarities Eucldean and nonEuclidean translations, but are beginning to suspect some differences. \end{document}