We check conformality of distances and lengths¶

Exercise I.11.24 [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}{⋋} $ Let the focal length with $\sigma=\rs=0$: \begin{equation} l_{f_{\bs}}^2(\lvec{PP'}) = -\bs p^2-2vp, \qquad \text{where} \quad p = \frac{(u'-u)^2%-\sigma v'^2 }{2(v'-v)}. \end{equation} Check that $l_{f_{\bs}}$ is p-conformal and, moreover, \begin{equation} \label{eq:parab-conf-factor} \lim_{v'\rightarrow \infty} \frac{l_{f_{\bs}}(\lvec{QQ'})}{l_{f_{\bs}}(\lvec{PP'})} =\frac{1}{(cu+d)^2},\qquad \text{where}\quad g= \begin{pmatrix} a&b\\c&d \end{pmatrix} \end{equation} and $Q=g\cdot P$, $Q'=g\cdot P'$.

Solution. We define the (\SL) identity for algebraic simplification.

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

Point P with the transformation gP.

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()]

Another point with its transformation.

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()]

The routine which is used to make the check.

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({sign : 0, u1 : x,v1 : y/t}));
    if (Len_cD.has(t)):
        Len_cD = Len_cD.series(relational(t, 0, eq), 1).op(0).normal()
    return ("This length is conformal: %s" %\
    (not(Len_cD.is_zero() or Len_cD.has(t) or Len_cD.has(x) or Len_cD.has(y))))

Use expressions for length verified in le-length-focus-exp.py

print("The length from parabolic focus is conformal in parabolic sense:")
sign5=0;
p=(pow(u1-u,2))/(v1-v)/2
check_conformality(-sign1*pow(p, 2) - 2*v*p)

The length from parabolic focus is conformal in parabolic sense:

'This length is conformal: True'

This notebook is a part of the MoebInv notebooks project [2] .

References¶

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.
Vladimir V. Kisil, MoebInv notebooks, 2019.

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