None ex.I.4.15
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Print explicit image of a cycle under conjugation

Exercise I.4.15 [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}{⋋} $ Check that the image $(\tilde{k},\tilde{l},\tilde{n},\tilde{m})$ of the cycle $(k,l,n,m)$ under similarity with $g= \begin{pmatrix} a&b\\c&d \end{pmatrix}\in\SL$ is \begin{equation} (\tilde{k},\tilde{l},\tilde{n},\tilde{m})= (k d^2+2 l c d+m c^2, k b d+l( bc+a d)+m a c, n, k b^2+2 l a b+m a^2). \end{equation} This can also be presented through matrix multiplication: \begin{equation} \begin{pmatrix} \tilde{k}\\\tilde{l}\\\tilde{n}\\\tilde{m} \end{pmatrix}= \begin{pmatrix} d^2&2 c d&0\ & c^2\\ b d\ &bc+a d\ &0\ & a c\\ 0&0&1\ &0\\ b^2&2 a b&0\ & a^2 \end{pmatrix} \begin{pmatrix} k\\l\\n\\m \end{pmatrix}. \end{equation}

Solution. A one-line check

In [1]:
from init_cycle import *
Latex(f"Cycle ${C.string()}$ is mapped to ${C.sl2_similarity(a,b,c,d,e,sign_mat).string()}$")

            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]:
Cycle $(k, {\left(\begin{array}{cc}{l}&n\end{array}\right)}^{{symbol50} }, m)$ is mapped to $(2 d {l} c+ m c^{2}+ d^{2} k, {\left(\begin{array}{cc} a m c+ d a {l}+ b {l} c+ d k b&- {( n b c- n d a)} s\end{array}\right)}_{{symbol205} }, k b^{2}+2 a b {l}+ a^{2} m)$

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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