None ex.I.6.38
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Cycle similarity of the real line

Exercise I.6.38 [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}} $ The image $\cycle[\hat]{s_1}{\bs} \realline{s}{\bs} \cycle[\hat]{s_1}{\bs}$ of the real line under inversion in $\cycle[\hat]{s_1}{\bs}=(k,l,n,m)$ with $s\neq 0$ is the cycle \begin{equation} (2 s s_1 \bs k n,\ 2 s s_1 \bs l n,\ -\det(\cycle{s_1}{\bs}),\ 2 s s_1 \bs m n). \end{equation} It is the real line again if $\det(\cycle{s_1}{\bs})\neq0$ and either

  1. $s_1 n=0$, in which case it is a composition of $\SL$-action by $ \begin{pmatrix} l&-m\\k &-l \end{pmatrix}$ and the reflection in the real line, or
  2. $\bs=0$, i.e. the parabolic case of the cycle space.

If either of two conditions is satisfied then f-orthogonality $\cycle[\tilde]{s}{\bs}\sperp \cycle{s}{\bs}$ is preserved by the $\bs$-similarity with $\cycle[\hat]{s_1}{\bs}$.

Solution. One-line calculation:

In [1]:
from init_cycle import *
Latex(f"Image of the real line is the cycle: ${real_line.cycle_similarity(C,es,sign_mat,sign_mat1).normal().string()}$")

            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
Out[1]:
Image of the real line is the cycle: $(2 \sigma_1 n s k s_1, {\left(\begin{array}{cc}2 \sigma_1 n s {l} s_1&- s m k+ s {l}^{2}+ \sigma_1 n^{2} s s_1^{2}\end{array}\right)}_{{symbol156} }, 2 \sigma_1 n s m s_1)$

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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