None ex.I.6.24.ii

Check that the image of zero-radius cycle is a zero-radius cycle

Exercise I.6.24.ii [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}} $ The image $\zcycle[\tilde]{s}{\sigma }=\cycle{s_2}{\sigma} \overline{\zcycle{s_1}{\sigma}}\cycle{s_2}{\sigma}$ of a $\sigma$-zero-radius cycle $\zcycle[\tilde]{s_1}{\sigma}$ under the cycle similarity is a $\sigma$-zero-radius cycle $\zcycle[\tilde]{s_1}{\sigma}$. The $(s_1s_2)$-centre of $\zcycle[\tilde]{s}{\bs}$ is the linear-fractional transformation of the $(s_2/s_1)$-centre of $\zcycle{s}{\bs}$.

Solution For the conjugated cycle Zn of a zero-radius cycle Z we check that that Zn is a zero-radius as well.

In [1]:
from init_cycle import *
Zn=cycle2D(Z.cycle_similarity(C,e,sign_mat1,sign_mat2,pow(sign_mat1,-1).evalm()))
"Conjugation of a zero-radius cycle is a zero-radius cycle: %s" % \
Zn.hdet(e,sign_mat1).normal().subs({pow(s1,2) : 1},subs_options.algebraic)\
.normal().is_zero()
            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
Out[1]:
'Conjugation of a zero-radius cycle is a zero-radius cycle: True'

P is the h-centre of the cycle Z.

In [2]:
P=Z.center(diag_matrix([-1,1]))

Then we calculate the library's cycle::mooebius_map() of P defined by FSCc matrix of a generic cycle C. It coincides with the linear-fractional transformation defined by this matrix explicitly.

In [3]:
PM=C.moebius_map(P, e, (sign_mat*sign_mat1).evalm())

For the Möbius transformation we are using the conjugated value PN of P, which is the elliptic centre.

In [4]:
PN=P[0] - I*P[1]
"Cycle generates Moebius map: %s" %\
(PM[0] + I*PM[1]-((l+I*s*s1*n)*PN-m)/(k*PN+(-l+I*s*s1*n))).\
subs({sign : -1}).normal().is_zero()
Out[4]:
'Cycle generates Moebius map: True'

Now, Pc is the $s_2 s_1$-centre of Zn and Pc1 is the linear-fractional map defined by the cycle C. For vector and paravector formalism the sign in calculations need to be adjusted.

In [5]:
Pc=Zn.center(diag_matrix([-1,-s2*s1]))
if (UseVectors):
    Pc1=C.moebius_map(Z.center(diag_matrix([-1,-s2/s1])))
else:
    Pc1=C.moebius_map(Z.center(diag_matrix([-1,s2/s1])))

Now we check the statement, that is Pc and Pc1 are the same for the non-zero values of s and s1.

In [6]:
"s1*s2-center of the similar cycle is the Moebius transformation of s2/s1-centre: %s" % \
((Pc1[0]-Pc[0]).normal().subs({pow(s1,2) : 1, pow(s2,2) : 1}, subs_options.algebraic).is_zero() \
 and (Pc1[1]-Pc[1]).normal().subs({pow(s1,2) : 1, pow(s2,2) : 1}, subs_options.algebraic).is_zero())
Out[6]:
's1*s2-center of the similar cycle is the Moebius transformation of s2/s1-centre: True'

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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