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Lemma: explicit expressions for lengths from center

Lemma I.7.11.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}{⋋} $ For two points $P=u+\alli v$, $P'=u'+\alli v'\in\Space[\sigma]{R}{}$, the $\bs$-length from the \CAS{le-length-centre-exp}% $\rs$-centre% \index{length!from centre}% \index{centre!length from} for $\rs=\pm1$ between $P$ and $P'$ is \begin{equation} \label{eq:k-center-point} l_{c_{\bs}}^2(\lvec{PP'}) = (u-u')^2-\sigma v'^2+2\rs v v' -\bs v^2. \end{equation}

Proof. We need one more clifford unit

In [1]:
from init_cycle import *
nu = varidx(symbol("nu"),2)
er = clifford_unit(nu, diag_matrix([-1, sign2]), 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

Now define a cycle passing P1...

In [2]:
C11 = C.subject_to([C.passing(P1), C.is_normalized()])

...and having its er-centre at P

In [3]:
C11 = C11.subject_to([C11.center(er)[0] == u, C11.center(er)[1]== v])

and output the result

In [4]:
Len=pow(u-u1,2)-sign*v1*v1+2*sign2*v*v1 -sign1*v*v
Latex("Length from *center* between $(u,v)$ and $(u1,v1)$ is $%s$: %s" %\
(Len, (Len-C11.radius_sq(es)).normal().subs({pow(sign2,2) : 1}, subs_options.algebraic).normal().is_zero()))
Out[4]:
Length from *center* between $(u,v)$ and $(u1,v1)$ is ${(u-{u'})}^{2}+2 {v'} v \sigma_2- \sigma_1 v^{2}- {v'}^{2} \sigma$: True

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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