Properties of conjugation of a real line to a certain cycle¶

Exercise I.6.31 [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}} $ Let $\cycle{s}{\bs}=(k, l, n,m)$ be a cycle such that $\bs \det\cycle{s}{\bs}>0$ for $\bs\neq 0$. Let us define the cycle $\cycle[\tilde]{s}{\bs}$ by the quadruple $(k, l, n\pm\sqrt{\bs \det\cycle{s}{\bs}},m)$. Then:

$\cycle[\tilde]{s}{\bs} \cycle{s}{\bs} \cycle[\tilde]{s}{\bs} = \Space{R}{}$ and $\cycle[\tilde]{s}{\bs} \Space{R}{} \cycle[\tilde]{s}{\bs} = \cycle{s}{\bs}$.

$\cycle[\tilde]{s}{\bs}$ and $\cycle{s}{\bs}$ have common

roots.

In the $\bs$-implementation, the cycle $\cycle{s}{\bs}$ passes the centre of $\cycle[\tilde]{s}{\bs}$.

Hint. One can directly observe for real roots, since they are fixed points of the inversion. Also, the transformation of $\cycle{s}{\bs}$ to a flat cycle implies that $\cycle{s}{\bs}$ passes the centre of inversion, hence. There is also a CAS calculation for this.

Solution. We independently check values $\sigma=-1$ and $\sigma=1$.

from init_cycle import *
for si in range(-1,2,2):
    print("Inversion to the real line (with %s sign):" % si)
    # We define a cycle by the required quadruple
    C9 = cycle2D(k, [l, n+si*pow(C.hdet(es)*sign1, half)], m, es)

    # Check that the conjugation send the real line to C and wise verse 
    print("   Conjugation of the real line is the cycle C: %s" %\
    real_line.cycle_similarity(C9, es).subs({pow(sign1,2) : 1}, subs_options.algebraic).is_equal(C))
    print("   Conjugation of the cycle C is the real line: %s" %\
    C.cycle_similarity(C9, es).subs({pow(sign1,2) : 1}, subs_options.algebraic).is_equal(real_line))

    # Check roots and the centre incidence
    print("   Inversion cycle has common roots with C: %s" %\
    (C9.val([C.roots()[0], 0]).numer().normal().is_zero()\
    and C9.val([C.roots()[1], 0]).numer().normal().is_zero()))
    print("   C passing the centre of inversion cycle: %s" %\
    cycle2D(C, es).val(C9.center()).numer().subs({sign1 : sign}).expand()\
          .subs({pow(sign,2) : 1}, subs_options.algebraic).is_zero())

            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
Inversion to the real line (with -1 sign):
   Conjugation of the real line is the cycle C: True
   Conjugation of the cycle C is the real line: True
   Inversion cycle has common roots with C: True
   C passing the centre of inversion cycle: True
Inversion to the real line (with 1 sign):
   Conjugation of the real line is the cycle C: True
   Conjugation of the cycle C is the real line: True
   Inversion cycle has common roots with C: True
   C passing the centre of inversion cycle: 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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