Conformality of lengths¶

$ \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}} $ Exervise I.7.14.ii [1] Show $\SL$-conformality of the lengths from centres \begin{equation} \label{eq:k-center-point} l_{c_{\bs}}^2(\lvec{PP'}) = (u-u')^2-\sigma v'^2+2\rs v v' -\bs v^2. \tag{I.7.6} \end{equation} any combination of values of $\sigma$, $\bs$ and $\rs$.

Solution. We define the simplification for matrices in $\SL$:

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 image of $P=(u,v)$ under the Möbius transformation and make the component-wise substitution of the simplification rule.

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()]

The same for the image of $P_1=(u_1,v_1)$ under the Moebius 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()]

Define the procedure to check the respective distances:

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.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 regularised expression for length checked in le-length-centre-exp.ipynb aka ex.I.7.11.i.ipynb.

print("Length from center")
Len_c=pow(u-u1,2)-sign*v1*v1+2*sign2*v*v1 -sign1*v*v
check_conformality(Len_c)

Length from center

'This length is conformal: True'

Use expressions for length verified in le-length-focus-exp.ipynb and ex.I.7.11.ii.ipynb with elliptic focus

print("Lengths from focus")
sign5=sign4
p=sign5*(-(v1-v)+pow(sign5*pow((u1-u), 2)+pow((v1-v), 2) -sign5*sign*pow(v1, 2),half))
check_conformality((sign5-sign1)*pow(p, 2) - 2*v*p)

Lengths from focus

'This length is conformal: True'

Now we check the similar equation for hyperbolic focus

sign5=sign4
p=sign5*(-(v1-v)-pow(sign5*pow((u1-u), 2)+pow((v1-v), 2) -sign5*sign*pow(v1, 2),half))
check_conformality((sign5-sign1)*pow(p, 2) - 2*v*p)

'This length is conformal: True'

Finally we check that length from parabolic focus is not conformal in usual sense

print("The length from parabolic focus is not conformal in usual sense (shall be False):")
sign5=0;
p=(pow(u1-u,2)-sign*pow(v1,2))/(v1-v)/2
check_conformality((sign5-sign1)*pow(p, 2) - 2*v*p)

The length from parabolic focus is not conformal in usual sense (shall be False):

'This length is conformal: False'

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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