None ex.I.7.26
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Infinitesimal radius cycle and orthogonality condition

Exercise I.7.26 [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 the infinitesimal cycle and $\cycle[\breve]{s}{\bs}=(k,l,n,m)$ be a generic cycle. Then:

  1. Both the orthogonality condition $\cycle{s}{\bs}\perp \cycle[\breve]{s}{\bs}$ and the f-orthogonality $\cycle[\breve]{s}{\bs} \sperp \cycle{s}{\bs}$ are given by \begin{equation} ku_0^2-2lu_0+m=O(\epsilon). \end{equation} In other words, the cycle $\cycle[\breve]{s}{\bs}$ has the real root $u_0$.
  2. The f-orthogonality $\cycle{s}{\bs} \sperp \cycle[\breve]{s}{\bs}$ is given by \begin{equation} \label{eq:f-orthog-infinites} ku_0^2-2lu_0-2nv_0+m=O(\epsilon). \end{equation} In other words, the cycle $\cycle[\breve]{s}{\bs}$ passes the focus $(u_0,v_0)$ of the infinitesimal cycle in the p-implementation.

Solution. Define the new symbol for vertical coordinate of the focus.

In [1]:
from init_cycle import *
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 procedure which will check all statesments.

In [2]:
def infinitesimal_ortho(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)
    display(Latex("Checking infinitesimal cycle: $%s$" % C10.string()))
    display(Latex("  Orthogonality (leading term) to infinitesimal cycle is: $%s$" %\
    C.is_orthogonal(C10, e).series(epsilon_is_zero, 1).normal()))
    display(Latex("  f-orthogonality of other cycle to infinitesimal: $%s$" %\
    C.is_f_orthogonal(C10, es).series(epsilon_is_zero, 1).normal()))
    return Latex("  f-orthogonality of infinitesimal cycle to other: $%s$" %\
    C10.is_f_orthogonal(C, es).series(epsilon_is_zero, 3).normal())

The first type of infinitesimal cycle

In [3]:
infinitesimal_ortho([n==(vp-(pow(pow(vp,2)-pow(epsilon,2)*(sign2-sign1),half)))/(sign2-sign1)])
Checking infinitesimal cycle: $(1, {\left(\begin{array}{cc}u&-\frac{vp-\sqrt{vp^{2}+ {(\sigma_1-\sigma_2)} \epsilon^{2}}}{\sigma_1-\sigma_2}\end{array}\right)}^{{symbol114} }, \epsilon^{2}+u^{2}-\frac{ \sigma_1 {(vp-\sqrt{vp^{2}+ {(\sigma_1-\sigma_2)} \epsilon^{2}})}^{2}}{{(\sigma_1-\sigma_2)}^{2}})$
Orthogonality (leading term) to infinitesimal cycle is: ${( u^{2} k-2 u {l}+m==0)}+\mathcal{O}(\epsilon)$
f-orthogonality of other cycle to infinitesimal: ${(-2 u n {l}+ u^{2} n k+ n m==0)}+\mathcal{O}(\epsilon)$
Out[3]:
f-orthogonality of infinitesimal cycle to other: ${(0==0)}+{(0==0)} \epsilon+{(\frac{1}{2} (\frac{ u^{2} k-2 u {l}-2 vp n+m}{vp}==0))} \epsilon^{2}+\mathcal{O}(\epsilon^{3})$

The parabolic-focus infinitesimal cycle.

In [4]:
infinitesimal_ortho([n==pow(epsilon,2)/2/vp, sign2==sign1])
Checking infinitesimal cycle: $(1, {\left(\begin{array}{cc}u&\frac{1}{2} \frac{\epsilon^{2}}{vp}\end{array}\right)}^{{symbol264} }, -\frac{1}{4} \frac{ \sigma_1 \epsilon^{4}}{vp^{2}}+\epsilon^{2}+u^{2})$
Orthogonality (leading term) to infinitesimal cycle is: ${( u^{2} k-2 u {l}+m==0)}+\mathcal{O}(\epsilon)$
f-orthogonality of other cycle to infinitesimal: ${(-2 u n {l}+ u^{2} n k+ n m==0)}+\mathcal{O}(\epsilon)$
Out[4]:
f-orthogonality of infinitesimal cycle to other: ${(0==0)}+{(0==0)} \epsilon+{(\frac{1}{2} (\frac{ u^{2} k-2 u {l}-2 vp n+m}{vp}==0))} \epsilon^{2}+\mathcal{O}(\epsilon^{3})$

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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