Transverse lines for K-orbits¶

Exercise I.3.13.iv [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 transverse line obtained from the vertical axis has the equations: \begin{equation} (u^2-\sigma v^2)+2\cot(2\phi) u-1=0, \quad \text{for } g=\begin{pmatrix} \cos \phi& \sin \phi\\ -\sin \phi\,& \cos \phi \end{pmatrix} \in K. \end{equation} Hint. A direct calculation for a point ((0,s)) in the formula is possible but demanding. A computer symbolic calculation is provided as well.

Solution. Create a point on the vertical axis

from init_cycle import *
P0=[0,v]

            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

Make its Moebius map with the matrix [a,b;-b,a] from subgroup K

[U,V]=clifford_moebius_map(sl2_clifford(a,b,-b,a,e),P0,e)

Verify the identity for the cycle

print("Point belongs to the cycle: %s" % \
bool(((U*U-sign*V*V)+(a*a-b*b)/a/b*U-1).normal()==0))

Point belongs to the cycle: True

Alternatively create this cycle...

Cp=cycle2D(1,[-(a*a-b*b)/a/b/2,0],-1,e)

...and check that it passes through the point

print("Cycle passes the point: %s" % bool(Cp.passing([U,V])))

Cycle passes the point: True

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

References¶

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.
Vladimir V. Kisil, MoebInv notebooks, 2019.

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