None ex.I.7.24
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Infinitesimal cycle is the locus of infinitesimally close points

Exercise I.7.24 [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 infinitesimal cycle consists of points which are infinitesimally close (in the sense of length from focus) to its focus $F=(u_0, v_0)$: \begin{equation} \label{eq:inf-cycle-points} % (u_0+\epsilon x, v_0+ v_0x^2-\rs \frac{\epsilon^2}{4v_0})\\ (u_0+\epsilon u, v_0+v_0u^2+((\bs-\rs) u^2-\rs) \frac{\epsilon^2}{4v_0}+O(\epsilon^3)). \end{equation}

Solution. Some algebraic substitutions for infinitesimal cycles.

In [1]:
from init_cycle import *
sign_cube = (pow(sign, 3) == sign)
sign1_cube = (pow(sign1, 3) == sign1)
vp=possymbol("vp")

            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 for the check

In [2]:
def infinitesimal_points(nval):
    C10 = cycle2D(1, [u, n],  pow(u,2)-pow(n,2)*sign1+pow(epsilon,2), e).subs(nval)
    display(Latex("Checking infinitesimal cycle: $%s$" % C10.string()))
    S=lsolve([C10.subs({sign : 0}).passing([u+epsilon*x, vp+y])], [y])[y].op(1).series(relational(epsilon, 0, eq), 3).normal()
    return Latex("  Infinitesimal cycle passing points $(u+%s, vp+%s)$" % (epsilon*x, S))

The first type of infinitesimal cycles

In [3]:
infinitesimal_points([n==(vp-(pow(pow(vp,2)-pow(epsilon,2)*(sign2-sign1),half)))/(sign2-sign1)])
Checking infinitesimal cycle: $(1, {\left(\begin{array}{cc}u&\frac{vp-\sqrt{- \epsilon^{2} {(\sigma_2-\sigma_1)}+vp^{2}}}{\sigma_2-\sigma_1}\end{array}\right)}^{{symbol114} }, u^{2}+\epsilon^{2}-\frac{ {(vp-\sqrt{- \epsilon^{2} {(\sigma_2-\sigma_1)}+vp^{2}})}^{2} \sigma_1}{{(\sigma_2-\sigma_1)}^{2}})$
Out[3]:
Infinitesimal cycle passing points $(u+ \epsilon x, vp+{( x^{2} \sigma_1^{2}-2 \sigma_2 x^{2} \sigma_1+ \sigma_2^{2} x^{2})} \epsilon^{2}+\mathcal{O}(\epsilon^{3}))$

The parabolic focus infinitesimal cycle

In [4]:
infinitesimal_points([n==pow(epsilon,2)/2/vp, sign2==sign1])
Checking infinitesimal cycle: $(1, {\left(\begin{array}{cc}u&\frac{1}{2} \frac{\epsilon^{2}}{vp}\end{array}\right)}^{{symbol123} }, u^{2}+\epsilon^{2}-\frac{1}{4} \frac{ \epsilon^{4} \sigma_1}{vp^{2}})$
Out[4]:
Infinitesimal cycle passing points $(u+ \epsilon x, vp+{(-4 vp^{2} x^{2})}+{(\sigma_1)} \epsilon^{2})$

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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