None ex.I.5.12.iii
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Cycle product for cycles from centres and radii

Exercise 5.12.iii [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 $\cycle{}{\bs}$ and $\cycle[\tilde]{}{\bs}$ be two cycles defined by e-centres $(u,v)$ and $(\tilde{u}, \tilde{v})$ with $\sigma$-determinants $-r^2$ and $-\tilde{r}^2$, respectively. Then, their $\bs$-cycle product explicitly is \begin{equation} \label{eq:cycle-prod-rad} \scalar{\cycle{}{\bs}}{\cycle[\tilde]{}{\bs}}= (u-\tilde{u})^2-\sigma(v-\tilde{v})^2-2(\sigma-\bs)v\tilde{v}-r^2-\tilde{r}^2. \end{equation}

Solution. A simple calculation.

In [1]:
from init_cycle import *
C3=cycle2D([u,v],e,pow(r,2))
C4=cycle2D([u1,v1],e,pow(r1,2))
Latex(f"Cycle product for cycles from centres and radii: ${C3.cycle_product(C4,es)}$")

            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]:
Cycle product for cycles from centres and radii: $-r^{2}+{u'}^{2}+u^{2}- \sigma v^{2}-r'^{2}+2 {v'} \sigma_1 v-2 u {u'}- \sigma {v'}^{2}$

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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