None ex.I.7.25
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Invariance of family of infinitesimal radius cycles under group and cycle conjugation

Exercise I.7.25 [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 image of (\SL)-action on an infinitesimal-radius cycle by conjugation is an infinitesimal-radius cycle of the same order.

The image of an infinitesimal cycle under cycle conjugation is an infinitesimal cycle of the same or lesser order.

Solution. Some common algebraic substitutions for infinitesimal radius cycles.

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

The routine to make all checks.

In [2]:
def infinitesimal_trans(nval):
    epsilon_is_zero = relational(epsilon, 0, eq)
    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()))
    display(Latex("  Image under SL2(R) of infinitesimal cycle has radius squared: $%s$" %\
    C11.radius_sq(es).subs(sl2_relation1, subs_options.algebraic).\
    subs(sign_cube, subs_options.algebraic).series(epsilon_is_zero, 3).normal()))
    return Latex("  Image under cycle similarity of infinitesimal cycle has radius squared: $%s$" %\
    C10.cycle_similarity(C, es).radius_sq(es).subs(sign_cube, subs_options.algebraic)\
    .series(epsilon_is_zero, 3).normal())

The check for the first type of infinitesimal cycles.

In [3]:
infinitesimal_trans({n : (vp-(pow(pow(vp,2)-pow(epsilon,2)*(sign2-sign1),half)))/(sign2-sign1)})
Checking infinitesimal cycle: $(1, {\left(\begin{array}{cc}u&-\frac{vp-\sqrt{vp^{2}+ {(\sigma_1-\sigma_2)} \epsilon^{2}}}{\sigma_1-\sigma_2}\end{array}\right)}^{{symbol114} }, -\frac{ {(vp-\sqrt{vp^{2}+ {(\sigma_1-\sigma_2)} \epsilon^{2}})}^{2} \sigma_1}{{(\sigma_1-\sigma_2)}^{2}}+u^{2}+\epsilon^{2})$
Image under SL2(R) of infinitesimal cycle has radius squared: ${(-\frac{1}{{( u^{2} c^{2}+2 u c d+d^{2})}^{2}})} \epsilon^{2}+\mathcal{O}(\epsilon^{3})$
Out[3]:
Image under cycle similarity of infinitesimal cycle has radius squared: ${(\frac{2 m k {l}^{2}-{l}^{4}+2 \sigma_1 n^{2} {l}^{2}- m^{2} k^{2}-2 \sigma_1 n^{2} m k- \sigma_1^{2} n^{4}}{{( u^{2} k^{2}+{l}^{2}-2 u k {l}- \sigma_1 n^{2})}^{2}})} \epsilon^{2}+\mathcal{O}(\epsilon^{3})$

The check for parabolic focus infinitesimal radius cycle.

In [4]:
infinitesimal_trans({n : pow(epsilon,2)/2/vp, sign2 : sign1})
Checking infinitesimal cycle: $(1, {\left(\begin{array}{cc}u&\frac{1}{2} \frac{\epsilon^{2}}{vp}\end{array}\right)}^{{symbol4711} }, u^{2}+\epsilon^{2}-\frac{1}{4} \frac{ \sigma_1 \epsilon^{4}}{vp^{2}})$
Image under SL2(R) of infinitesimal cycle has radius squared: ${(-\frac{1}{{( u^{2} c^{2}+2 u c d+d^{2})}^{2}})} \epsilon^{2}+\mathcal{O}(\epsilon^{3})$
Out[4]:
Image under cycle similarity of infinitesimal cycle has radius squared: ${(\frac{2 m k {l}^{2}-{l}^{4}+2 \sigma_1 n^{2} {l}^{2}- m^{2} k^{2}-2 \sigma_1 n^{2} m k- \sigma_1^{2} n^{4}}{{( u^{2} k^{2}+{l}^{2}-2 u k {l}- \sigma_1 n^{2})}^{2}})} \epsilon^{2}+\mathcal{O}(\epsilon^{3})$

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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