Proposition about f-ghost cycle¶

Proposition I.6.40 [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}{⋋} $ Let $\cycle{s}{\bs}$ be a cycle. Then, its {f-ghost cycle} $\cycle[\breve]{\bs}{\bs} = \cycle{\chi(\sigma)}{\bs} \Space[\bs]{R}{\bs} \cycle{\chi(\sigma)}{\bs}$ is the reflection of the real line in $\cycle{\chi(\sigma)}{\bs}$, where $\chi(\sigma)$ is the {Heaviside function}. Then:

Cycles $\cycle{1}{\bs}$ and $\cycle[\breve]{\bs}{\bs}$ have the same roots.
The $\chi(\sigma)$-centre of $\cycle[\breve]{\bs}{\bs}$ coincides with the ($-\bs$)-focus of $\cycle{s}{\bs}$, consequently all straight lines $\bs$-f-orthogonal to $\cycle{s}{\bs}$ pass its ($-\bs$)-focus.
f-inversion in $\cycle{s}{\bs}$ defined from f-orthogonality (see Definition~\ref{item:inversion-orthogonality}) coincides with the usual inversion in $\cycle[\breve]{\bs}{\bs}$.

Proof. We construct the cycle C6 f-orthogonal to C and passing P

from init_cycle import *
C6=C1.subject_to([C1.passing(P), C.is_f_orthogonal(C1, es)])

            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

A straight line C7 with the above properties

C7=C6.subject_to(C6.is_linear())

The point with the first coordinate of the centre of C which the line passes

u4 = C.center()[0]
v4=C7.roots(u4, False)[0].normal()
"All f-orthogonal lines come through the bs-focus of cycle: %s" %\
(C.focus(diag_matrix([-1, -sign1]), True)-matrix([[u4], [v4]])).normal().is_zero()

'All f-orthogonal lines come through the bs-focus of cycle: True'

We build the ghost cycle C8

C8=real_line.cycle_similarity(C, es, diag_matrix([1, sign1]), diag_matrix([1, jump_fnct(sign)]), diag_matrix([1, sign1])).normalize(n*k)
R=C.roots()
"f-Ghost cycle has common roots with C: %s" % \
(C8.val([R[0], 0]).numer().normal().is_zero() and C8.val([R[1], 0]).numer().normal().is_zero())

'f-Ghost cycle has common roots with C: True'

Latex("$H(\sigma)$-center of the f-ghost cycle coincides with $\breve{\sigma}$-focus of C: %s" %\
(C8.center(diag_matrix([-1,jump_fnct(sign)]), True)\
-C.focus(diag_matrix([-1, -sign1]), True)).evalm().normal().is_zero_matrix())

Different formulae for vector and paravector formalism.

if (UseVectors):
    P3=C8.moebius_map(P,e,diag_matrix([1,-jump_fnct(sign)]))
else:
    P3=C8.moebius_map([P[0],-P[1]],e,diag_matrix([1,-jump_fnct(sign)]))
if (not UseVectors):
    P3=C8.moebius_map([P[0],-P[1]],e,diag_matrix([1,-jump_fnct(sign)]))
    # or complex conjugationof the vector case point point 
    #P3=[P3[0],-P3[1]]

"f-inversion in C coincides with inversion in f-ghost cycle: %s " %\
C6.val(P3).normal().is_zero()

'f-inversion in C coincides with inversion in f-ghost 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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