None th.I.4.13
Open In Colab Open In Colab Open In Colab Code Ocean

Google CoLab software installation

Proving conjugation formula for Fillmore-Springer-Cnops construction

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

Theorem I.4.13. [1] Let a matrix $M= \begin{pmatrix} a&b\\c&d \end{pmatrix}\in SL_2(\mathbb{R})$ define the Möbius transformation: \begin{equation} \begin{pmatrix} a&b\\c&d \end{pmatrix}:\ w\mapsto \frac{aw+b}{c w+d}, \tag{I.3.24}\label{eq:moebius-def} \end{equation} where $w=u+\iota v$ with $\iota^2=\sigma$.

The image $\tilde{C}^{s}_{\breve{\sigma}}$ of a cycle ${C}^{s}_{\breve{\sigma}}$ under transformation \eqref{eq:moebius-def} in $\mathbb{R}^2_{\sigma}$ is given by similarity of the matrix: \begin{equation} \label{eq:cycle-transform-short} \tilde{C}^{s}_{\breve{\sigma}} = M {C}^{s}_{\breve{\sigma}} M^{-1}. \end{equation} Explicitly, it means: \begin{equation} \label{eq:cycle-similarity-expl} \begin{pmatrix} \tilde{l}+\breve{\iota} s \tilde{n}&-\tilde{m}\\\tilde{k}&-\tilde{l}+\breve{\iota} s \tilde{n} \end{pmatrix} = \begin{pmatrix} a&b\\c&d \end{pmatrix} \begin{pmatrix} l+\breve{\iota} s n&-m\\k&-l+\breve{\iota} s n \end{pmatrix} \begin{pmatrix} d&-b\\-c&a \end{pmatrix}. \end{equation}

Proof. The statement can be fully demonstrated by symbolic computations in MoebInv. Here we will not use the file init_cycle.py and will redefine many objects from it. First, we need to introduce two generic metrics: one is for the point space and another is for the cycle space.

In [1]:
from cycle import *
sign = realsymbol('si', r'\sigma') # symbol to encode point space geometry
sign1=realsymbol("bs", r"\sigma_1")  # symbol to encode cycle space geometry
mu=idx(symbol("mu"),1) # index for a cliffor unit
nu=idx(symbol("nu"),1) # index for a cliffor unit
e=clifford_unit(mu,diag_matrix([sign])) # clifford unit for the point space
es=clifford_unit(nu,diag_matrix([sign1])) # clifford unit for the cycle space

            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.
     ---------------------------------------------

Now we define a generic cycle C by arbitrary coefficients $(k,l,n,m)$.

In [2]:
k = realsymbol("k") # Define a real variable k
l = realsymbol("l") # Define a real variable l
n = realsymbol("n") # Define a real variable n
m = realsymbol("m") # Define a real variable m
C = cycle2D(k, [l, n], m, e) # Define the cycle

Then, for a given point $(u,v)$ we build a cycle passing it. This is done by subject_to() menthod:

In [3]:
u = realsymbol('u') # the first coordinate of a point
v = realsymbol('v') # the second coordinate of the point
P = matrix([[u, v]]) # a point on the plane
C2=C.subject_to(C.passing(P))
Latex(f'Cycle C2=${C2}$ passes point $P={P}$: {bool(C2.passing(P))}')
Out[3]:
Cycle C2=$(k, {\left(\begin{array}{cc}l&n\end{array}\right)}^{{symbol14} }, \sigma v^{2} k- u^{2} k+2 n v+2 u l)$ passes point $P=\left(\begin{array}{cc}u&v\end{array}\right)$: True

Now we define an element of $SL_2(\mathbb{R})$ and construct the conjugation of the cycle C2 by similarity in the matrix $\begin{pmatrix}a&b\\c&d\end{pmatrix}$.

In [4]:
a = realsymbol('a')
b = realsymbol('b')
c = realsymbol('c')
d = realsymbol('d')
s = realsymbol('s')
C3=C2.sl2_similarity(a, b, c, d, es, matrix([[1,0],[0,s]]), True, matrix([[1,0],[0,1/s]])) 
Latex(f'The transformed cycle: C3=${C3}$')
Out[4]:
The transformed cycle: C3=$( \sigma v^{2} k c^{2}+2 u c^{2} l+2 d c l- u^{2} k c^{2}+ d^{2} k+2 n v c^{2}, {\left(\begin{array}{cc}2 u a c l+ d a l+2 n a v c+ \sigma a v^{2} k c- u^{2} a k c+ b c l+ d b k& s {(\frac{ d n a}{s}-\frac{ n b c}{s})}\end{array}\right)}_{{symbol90} }, 2 u a^{2} l- u^{2} a^{2} k+ b^{2} k+ \sigma a^{2} v^{2} k+2 n a^{2} v+2 a b l)$

This explicit form of the cycle may be not particularly enlightening. But we are moving on and define the Mobius map of the point $P$:

In [5]:
P1=clifford_moebius_map(sl2_clifford(a,b,c,d,e),P,e)
Latex(f'${P1}$')
Out[5]:
$\left(\begin{array}{cc}\frac{ u^{2} a c}{2 u d c+ u^{2} c^{2}+d^{2}- \sigma v^{2} c^{2}}+\frac{ d b}{2 u d c+ u^{2} c^{2}+d^{2}- \sigma v^{2} c^{2}}+\frac{ u d a}{2 u d c+ u^{2} c^{2}+d^{2}- \sigma v^{2} c^{2}}-\frac{ \sigma a v^{2} c}{2 u d c+ u^{2} c^{2}+d^{2}- \sigma v^{2} c^{2}}+\frac{ u b c}{2 u d c+ u^{2} c^{2}+d^{2}- \sigma v^{2} c^{2}}&-\frac{ v b c}{2 u d c+ u^{2} c^{2}+d^{2}- \sigma v^{2} c^{2}}+\frac{ d a v}{2 u d c+ u^{2} c^{2}+d^{2}- \sigma v^{2} c^{2}}\end{array}\right)$

The two last outputs are cumbersome, fortunately computer can make the final check for us. We only need to suggest that the $SL_2(\mathbb{R})$ substitution $a=(1+bc)/d$ shall be used for simplification. Also we assume that $s^2=1$.

In [6]:
'Conjugated cycle C3 passes the Moebius image P1 of P: %s' % \
C3.val(P1).subs({a : (1+b*c)/d, pow(s,2) : 1}).normal().is_zero()
Out[6]:
'Conjugated cycle C3 passes the Moebius image P1 of P: True'

This finishes the proof by a direct calculation. $\Box$

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.

Back to Folder