None ex.I.6.33
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Check Yaglom inversion properties

Exercise I.6.33 [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}} \newcommand{\sperp}{⋋} $ The inversion of the second kind is a composition of three Möbius transformations defined by cycles $(1,l,2m,l^2+m/k)$, $(1,l,0,l^2+m/k)$ and the real line in the parabolic point space $\Space[p]{R}{}$.

Solution. Two additional cycles are defined.

In [1]:
from init_cycle import *
C2=cycle2D([l, 0], e, -m/k)
C3=cycle2D([l, 2*m], e, -m/k)

            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
In [2]:
Latex("Consider composition of three reflections in two cycles $%s$ and $%s$ and the real line" %\
(C3.subs({sign : 0}).normal().string(), C2.subs({sign : 0}).normal().string()))
Out[2]:
Consider composition of three reflections in two cycles $(1, {\left(\begin{array}{cc}{l}&2 m\end{array}\right)}^{{symbol128} }, \frac{m+ k {l}^{2}}{k})$ and $(1, {\left(\begin{array}{cc}{l}&0\end{array}\right)}^{{symbol114} }, \frac{m+ k {l}^{2}}{k})$ and the real line

Different formulae are used for vectors and paravector formalisms

In [3]:
if (UseVectors):
    P0=matrix([[u,v]])
else:
    P0=matrix([[u,-v]])
    
Latex("The composition is equal to Yaglom inversion of the second kind: %s" %\
(real_line.moebius_map(\
C2.moebius_map(\
C3.moebius_map(P0))).subs(sign==0)\
-matrix([[u, 2*(k*pow(u-l,2)+m)-v]])).normal().is_zero())

Warning: it is safer to avoid substitutions from relational, use Python dictionaries instead
Out[3]:
The composition is equal to Yaglom inversion of the second kind: True

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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