None ex.I.7.8
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Check the formula for an inversive distance for a generic cycle

Exercise I.7.8 [1] $ \newcommand{\scalar}[3][]{\left\langle #2,#3 \right\rangle_{#1}} \newcommand{\Space}[3][]{\mathbb{#2}^{#3}_{#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}} \newcommand{\rmi}{\mathrm{i}} \newcommand{\alli}{\iota} \newcommand{\rme}{\mathrm{e}} \newcommand{\rmd}{\mathrm{d}} \newcommand{\rmh}{\mathrm{j}} \newcommand{\rmp}{\varepsilon} \newcommand{\modulus}[2][]{\left| #2 \right|_{#1}} \newcommand{\sperp}{⋋} $ Let two cycles have e-centres $P$ and $P'$ with $\bs$-radii $r_{\bs}$ and $r_{\bs}'$. Then, the $(\bs,\bs)$-power% \index{power!cycle, of}% \index{point!cycle, of} of one cycle with respect to another is, \begin{equation} \frac{d_{\bs,\bs}^2(P,P')-r_{\bs}^2-r_{\bs}^{\prime 2}}{2r_{\bs}\, r_{\bs}'}. \end{equation}

Solution Define cycles C2 and C3 with radii r and r1 respectively

In [1]:
from init_cycle import *
r=possymbol("r")
r1=possymbol("r1", "r_1")
C2=cycle2D([u,v],es,pow(r,2)).normalize_det()
C3=cycle2D([u1,v1],es,pow(r1,2)).normalize_det()

            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
In [2]:
Latex(f"Cycle C2 is: ${C2.string()}$")
Out[2]:
Cycle C2 is: $(\frac{1}{r}, {\left(\begin{array}{cc}\frac{u}{r}&\frac{v}{r}\end{array}\right)}^{{symbol116} }, -\frac{r^{2}+ \sigma_1 v^{2}-u^{2}}{r})$
In [3]:
Latex("Power of a cycle with respect to another: $%s$" % \
C2.cycle_product(C3,e).normal())
Out[3]:
Power of a cycle with respect to another: $\frac{2 \sigma {v'} v-r^{2}- \sigma_1 v^{2}-r_1^{2}+u^{2}- {v'}^{2} \sigma_1-2 u {u'}+{u'}^{2}}{ r r_1}$

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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