None ex.I.5.23.i
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Check the formula for an inversive distance for circles

Exercise I.5.23.i [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}{⋋} $ Show that the (e,e)-power of a circle $\cycle{}{}$ with respect to another circle $\cycle[\tilde]{}{}$ has an absolute value equal to the \emph{inversive distance} between $\cycle{}{}$ and $\cycle[\tilde]{}{}$, see [Beardon95, Defn.~3.2.2] [Beardon07a, S~4] [CoxeterGreitzer, S~5.8 and Thm.~5.91]: \begin{equation} d(\cycle{}{}, \cycle[\tilde]{}{})=\modulus{\frac{\modulus{c_1-c_2}^2-r_1^2-r_2^2}{2r_1r_2}}. \end{equation} Here, $c_{1,2}$ and $r_{1,2}$ are the e-centres and radii of the circles. For other two cases, Exercise I.7.8 aka ex-power-cycle-exp.ipynb.

Solution. We define cycles C2 and C3 with radii r and r2.

In [1]:
from init_cycle import *
r=possymbol("r")
r1=possymbol("r1", "r_1")
C2=cycle2D([u,v],e,pow(r,2)).normalize_det()
C3=cycle2D([u1,v1],e,pow(r1,2)).normalize_det()
Latex("Inversive distance for circles is $%s$" % \
(C2.cycle_product(C3).subs({sign : -1}).normal()/2))

            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]:
Inversive distance for circles is $\frac{1}{2} \frac{{u'}^{2}-r^{2}+v^{2}-2 v {v'}-2 {u'} u+{v'}^{2}+u^{2}-r_1^{2}}{ r r_1}$

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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