None ex.I.10.14
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Check geodesics under the cayley transform

Exercise I.10.14 [1] $ \newcommand{\scalar}[3][]{\left\langle #2,#3 \right\rangle_{#1}} \newcommand{\Space}[3][]{\mathbb{#2}^{#3}_{#1}} \newcommand{\Space}[3][]{\mathbb{#2}^{#3}_{#1}} \newcommand{\cycle}[3][]{{#1 C^{#2}_{#3}}} \newcommand{\realline}[3][]{#1 R^{#2}_{#3}} \newcommand{\bs}{\breve{\sigma}} \newcommand{\zcycle}[3][]{#1 Z^{#2}_{#3}} \newcommand{\SL}[1][2]{\mathrm{SL}_{#1}(\Space{R}{})} \newcommand{\rs}{\sigma_r} \newcommand{\lvec}[1]{\overrightarrow{#1}} \newcommand{\rmi}{\mathrm{i}} \newcommand{\alli}{\iota} \newcommand{\rme}{\mathrm{e}} \newcommand{\rmd}{\mathrm{d}} \newcommand{\rmh}{\mathrm{j}} \newcommand{\rmp}{\varepsilon} \newcommand{\modulus}[2][]{\left| #2 \right|_{#1}} $ Check that:

  1. (e,h) Elliptic and hyperbolic Cayley transforms send the respective geodesics passing $\alli$ to the straight line passing the origin. Consequently, any geodesics is a cycle orthogonal to the boundary of the unit disk. The respective invariant metrics on the unit disks are, cf. [Yaglom79, Table~VI], \begin{equation} \sin_{\rs}^{-1} \frac{|w-w'|_\sigma}{2 \sqrt{(1+\sigma w\bar{w})(1+\sigma w'\bar{w}')}}, \end{equation} where $\rs=1$ in the elliptic case and has a value depending on the degree of space- or light-likeness in the hyperbolic case.
  2. (p) The $\bs$-parabolic Cayley transform maps the $\rs$-geodesics passing $\rmp$ to the parabolas passing the origin: \begin{equation} (\rs-4\bs+4t^2)u^2-8tu-4v=0. \end{equation} The invariant $\rs$-metric on the $\bs$-parabolic unit circle is \begin{equation} \sin_{\rs }^{-1} \frac{|u-u'|}{\sqrt{(1+v+\bs u^2)(1+v'+\bs u'^2)} }. \end{equation}

Solution. Make the check for elliptic and hyperbolic geodesics from $\rmi$

In [1]:
from init_cycle import *
Ceh=cycle2D(1,[t,0],sign,e)
print("Elliptic and hyperbolic geodesic on unit disk from the centre are straight lines: %s" %\
bool(cayley(Ceh,sign,0).subs(pow(sign,2)==1,subs_options.algebraic).is_linear()))

            Python wrappers for MoibInv Library
     ---------------------------------------------
Please cite this software as
V.V. Kisil, MoebInv: C++ libraries for manipulations in non-Euclidean geometry, SoftwareX, 11(2020),100385. doi:10.1016/j.softx.2019.100385.
     ---------------------------------------------

Using vector formalism and idx
Elliptic and hyperbolic geodesic on unit disk from the centre are straight lines: True

The parabolic geodesics are obtained by the rotation of the main one

In [2]:
Cp=cycle2D(sign2,[0,2],4,e)
Latex(f"${Cp.sl2_similarity([1,0,t,1],e).string()}$")
Out[2]:
$(4 t^{2}+\sigma_2, {\left(\begin{array}{cc}4 t&2\end{array}\right)}_{{symbol244} }, 4)$
In [3]:
Latex("Geodesics under the parabolic Cayley transform: $%s$" % \
cayley(Cp.sl2_similarity([1,0,t,1],e),0,sign1).string())
Out[3]:
Geodesics under the parabolic Cayley transform: $(-4 \sigma_1+4 t^{2}+\sigma_2, {\left(\begin{array}{cc}4 t&2\end{array}\right)}^{{symbol292} }, 0)$
In [4]:
Latex("Geodesics under the parabolic Cayley transform second form: $%s$" % \
cayley(Cp.sl2_similarity([1,0,t,1],e),0,sign1).string())
Out[4]:
Geodesics under the parabolic Cayley transform second form: $(-4 \sigma_1+4 t^{2}+\sigma_2, {\left(\begin{array}{cc}4 t&2\end{array}\right)}^{{symbol337} }, 0)$

This notebook is a part of the MoebInv notebooks project [2] .

References

  1. Vladimir V. Kisil. Geometry of Möbius Transformations: Elliptic, Parabolic and Hyperbolic Actions of $SL_2(\mathbb{R})$. Imperial College Press, London, 2012. Includes a live DVD.

  2. Vladimir V. Kisil, MoebInv notebooks, 2019.

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