None ex.I.7.29
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Focus of an infinitesimal cycle is mapped by SL(2,R) to the (almost) focus of image

Exercise I.7.29 [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}} $ Let $\cycle[\breve]{s}{\bs}$ be the image under $g\in\SL$ of an infinitesimal cycle $\cycle{s}{\bs}$. Then, the $\rs$-focus of $\cycle[\breve]{s}{\bs}$ is displaced from $g(u_0,v_0)$ by infinitesimals of order $\epsilon^2$, while both cycles have $\bs$-radius of order $\epsilon$.

Consequently, $\SL$-action is infinitesimally-conformal with respect to the length from the focus for all combinations of $\sigma$, $\bs$ and $\rs$.

Solution. We define few relations which will be used for algebraic simplification later.

In [1]:
from init_cycle import *
sign_cube = {pow(sign, 3) : sign}
sign1_cube = {pow(sign1, 3) : sign1}
vp=possymbol("vp")
sl2_relation = {c*b : a*d-1}
sl2_relation1 = {a : (1+b*c)/d}

            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

Here is a routine which will verify two statements from the Exercise.

In [2]:
def infinitesimal_trans(nval):
    C10 = cycle2D(1, [u, n],  pow(u,2)-pow(n,2)*sign1+pow(epsilon,2), e).subs(nval)
    C11=cycle2D(C10.sl2_similarity(a, b, c, d, et))
    display(Latex("Checking infinitesimal cycle: $%s$" % C10.string()))
    gP=clifford_moebius_map(sl2_clifford(a, b, c, d, et), matrix([[u],[vp]]), et)\
    .subs(sl2_relation1,subs_options.algebraic).normal()
    displ = (C11.focus(er, True).subs(nval) - gP).evalm()
    return Latex("Focus of the transormed cycle is displaced from transformation of focus by: $%s$" %\
    displ.subs(sl2_relation, subs_options.algebraic)\
    .subs({sign : 0,sign3 : 0,a : (1+b*c)/d}).series(relational(epsilon, 0, eq), 2).normal())

Checking for the first type of infinitesimal cycle

In [3]:
infinitesimal_trans({n : (vp-(pow(pow(vp,2)-pow(epsilon,2)*(sign2-sign1),half)))/(sign2-sign1)})
Checking infinitesimal cycle: $(1, {\left(\begin{array}{cc}u&-\frac{vp-\sqrt{vp^{2}+ {(\sigma_1-\sigma_2)} \epsilon^{2}}}{\sigma_1-\sigma_2}\end{array}\right)}^{{symbol114} }, -\frac{ \sigma_1 {(vp-\sqrt{vp^{2}+ {(\sigma_1-\sigma_2)} \epsilon^{2}})}^{2}}{{(\sigma_1-\sigma_2)}^{2}}+u^{2}+\epsilon^{2})$
Out[3]:
Focus of the transormed cycle is displaced from transformation of focus by: ${(\left(\begin{array}{c}0\\0\end{array}\right))}+{(\left(\begin{array}{c}0\\0\end{array}\right))} \epsilon+\mathcal{O}(\epsilon^{2})$

The parabolic focus infinitesimal cycle

In [4]:
infinitesimal_trans({n : pow(epsilon,2)/2/vp, sign2 : sign1})
Checking infinitesimal cycle: $(1, {\left(\begin{array}{cc}u&\frac{1}{2} \frac{\epsilon^{2}}{vp}\end{array}\right)}^{{symbol1243} }, -\frac{1}{4} \frac{ \sigma_1 \epsilon^{4}}{vp^{2}}+u^{2}+\epsilon^{2})$
Out[4]:
Focus of the transormed cycle is displaced from transformation of focus by: ${(\left(\begin{array}{c}0\\0\end{array}\right))}+{(\left(\begin{array}{c}0\\0\end{array}\right))} \epsilon+\mathcal{O}(\epsilon^{2})$

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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