None ex.I.7.30
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We check (the absence of) infinitesimal conformality of parabolic lengths

Exercise I.7.30 [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}{⋋} $ Show that, for lengths from foci and $\rs= 0$, the limit at point $y_0=u_0+\alli v_0$ does exist but depends on the direction $y=u+\alli v$: \begin{equation} \label{eq:focal-length-factor} \lim_{t\rightarrow 0} \frac{d(g\cdot y_0, g\cdot(y_0+ty))}{d(y_0, y_0+ty)}=\frac{1}{(d+cu_0)^2+\sigma c^2 v_0^2 -2 K c v_0(d+c u_0)}, \end{equation} where $ K=\frac{u}{v}$ and $g= \begin{pmatrix} a&b\\c&d \end{pmatrix}$. Thus, the length is not conformal.

Solution. We use the $\SL$ relation fro algebraic subctitutions.

In [1]:
from init_cycle import *
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

Define the first transformed point

In [2]:
gP=clifford_moebius_map(sl2_clifford(a, b, c, d, e), P, e)
gP=[gP[0].subs(sl2_relation1,subs_options.algebraic).normal(),\
gP[1].subs(sl2_relation1,subs_options.algebraic).normal()]

Define the second transformed point

In [3]:
gP1=clifford_moebius_map(sl2_clifford(a, b, c, d, e), P1, e)
gP1=[gP1[0].subs(sl2_relation1,subs_options.algebraic).normal(),\
gP1[1].subs(sl2_relation1,subs_options.algebraic).normal()]
t=possymbol("t")

The procedure to check the relations.

In [4]:
def check_conformality(Len_c):
    Len_cD= ((Len_c.subs({u : gP[0], v : gP[1], u1 : gP1[0], v1 : gP1[1]})\
    /Len_c).subs({u1 : u+t*x, v1 : v+t*y}))
    
    if (Len_cD.has(t)):
        Len_cD = Len_cD.normal().series(relational(t, 0, eq), 1).op(0).normal()
    return Latex("The factor is: $%s$" % Len_cD.normal())
In [5]:
print("The length from parabolic focus is not conformal in usual sense:")
p=(pow(u1-u,2)-sign*pow(v1,2))/(v1-v)/2
check_conformality(-sign1*pow(p, 2) - 2*v*p)

The length from parabolic focus is not conformal in usual sense:
Out[5]:
The factor is: $\frac{y^{2}}{{(2 c x v d- u^{2} c^{2} y- \sigma c^{2} v^{2} y+2 u c^{2} x v- d^{2} y-2 u c d y)}^{2}}$

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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