None ex.I.10.9

We verify the expression of the parabolic Cayley transform on the cycles

Exercise I.10.9 [1] Let $\newcommand{\Space}[3][]{\mathbb{#2}^{#3}_{#1}} \newcommand{\cycle}[3][]{{#1 C^{#2}_{#3}}} \newcommand{\realline}[3][]{#1 R^{#2}_{#3}} \newcommand{\bs}{\breve{\sigma}}$ $\cycle{s}{a}$ be a cycle in $\Space[\sigma]{R}{}$. Check that:

  • (e,h) In the elliptic or hyperbolic cases, the Cayley transform of the cycle $\cycle{}{\sigma}$ is $\realline{}{\sigma}\cycle[\hat]{}{\sigma }\cycle{}{\sigma}\cycle[\hat]{}{\sigma }\realline{}{\sigma}$, i.e. the composition of the similarity (I.6.10) by the cycle $\cycle[\hat]{s}{\sigma }=(\sigma ,0,1,1)$ and the similarity by the real line (see the first and last drawings in Fig.I.10.2.

  • (p) In the parabolic case, the Cayley transform maps a cycle $(k,l,n,m)$ to the cycle $(k-2\bs n, l, n ,m-2 n)$.

Hint We can follow a similar path to the proof of Theorem I.4.13. Alternatively, for the first part, we notice that the matrix $Y_\sigma $ of the Cayley transform and the FSCc matrix of the cycle $\cycle[\hat]{s}{\sigma }$ are different by a constant factor. The reflection in the real line compensates the effect of complex conjugation in the similarity (I.6.10).

Solution. Take the Cayley transform of a point...

In [1]:
from init_cycle import *
P0=(u+I*v-I)/(-I*(u+I*v)+1)
            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

... and its Möbius transformation

In [2]:
Pc=clifford_moebius_map(one, -e1, sign1*e1, one, P, e)

Compare both outputs:

In [3]:
print("Clifford calculations coinsides with complex numbers: %s" %\
((P0+P0.conjugate()-2*Pc[0]).subs({sign : -1,sign1 : -1}).normal().is_zero() and
(P0-P0.conjugate()-2*I*Pc[1]).subs({sign : -1,sign1 : -1}).normal().is_zero()))
Clifford calculations coinsides with complex numbers: 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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