None ex.I.3.7
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Check formulas for Moebius action

Exercise I.3.7 [1] Use CAS to check the formula for all $\sigma=-1$, $0$, $1$: \begin{equation} \label{eq:moebius-cumbersome} \begin{pmatrix} a&b\\c&d \end{pmatrix}:\ (u,v)\mapsto \left(\frac{(au+b)(c u+d) -\sigma cav^2}{( c u+d)^2 -\sigma (cv)^2}, \frac{v}{( c u+d)^2 -\sigma (cv)^2}\right). \tag{I.3.23} \end{equation}

Solution. Use map s (3.13) from the book [1]

In [1]:
from init_cycle import *
def s(P):
    return matrix([[pow(P[1],half),P[0]/pow(P[1],half)],[0,pow(P[1],-half)]])

            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

Also define map p (3.14), (3.17), (3.20) from [1]

In [2]:
def p(M,sig):
    return [(M.op(1)*M.op(3)-sig*M.op(0)*M.op(2))/(M.op(3)*M.op(3)-sig*M.op(2)*M.op(2)),
            1/(M.op(3)*M.op(3)-sig*M.op(2)*M.op(2))]

Here is the three different types of the non-conjugated subgroups

In [3]:
subgr=["K", "N", "A"]

Now we ready to output results of calculation:

In [4]:
result = []
for i in range(3):
    P1=p((matrix([[a,b],[c,d]])*s([u,v])).evalm(), i-1)
    display(Latex(f"For subgroup {subgr[i]} action is: ${P1[0].normal()}$, ${P1[1].normal()}$"))
For subgroup K action is: $\frac{ c a u^{2}+ c v^{2} a+ c b u+ d b+ d a u}{d^{2}+ c^{2} v^{2}+ c^{2} u^{2}+2 c d u}$, $\frac{v}{d^{2}+ c^{2} v^{2}+ c^{2} u^{2}+2 c d u}$
For subgroup N action is: $\frac{b+ a u}{d+ c u}$, $\frac{v}{{(d+ c u)}^{2}}$
For subgroup A action is: $\frac{ c a u^{2}- c v^{2} a+ c b u+ d b+ d a u}{d^{2}- c^{2} v^{2}+ c^{2} u^{2}+2 c d u}$, $\frac{v}{d^{2}- c^{2} v^{2}+ c^{2} u^{2}+2 c d u}$

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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