None ex.I.7.3
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Check basic formulae for determinant

Exercise I.7.3 [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}{⋋} $ Check the following relations:

  1. The $\bs$-radius of a cycle $\cycle{s}{\bs}$ is equal to $1/k$, where $k$ is the $(2,1)$-entry of the $\det$-normalised FSCc matrix of the cycle.
  2. Let $u_{\bs}$ be the second coordinate of a cycle's $\bs$-focus and $f$ be its focal length. Then, the square of the cycle's $\bs$-radius is \begin{equation} r^2_{\bs}=-4fu_{\bs}. \end{equation}
  3. Cycles have zero $\bs$-radius.

Solution. A sequence of the direct checks

In [1]:
from init_cycle import *
print("Radius of a cycle is equal to 1/k of det-normalised cycle:")
(1-C.radius_sq()*pow(C.normalize_det().get_k(),2)).normal().is_zero()

            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
Radius of a cycle is equal to 1/k of det-normalised cycle:
Out[1]:
True
In [2]:
print("Square of radius is minus four times focal length times v component of focus:")
(C.radius_sq(es)+4*C.focal_length()*C.focus(es)[1]).normal().is_zero()

Square of radius is minus four times focal length times v component of focus:
Out[2]:
True
In [3]:
print("Cycle (1,u,v,u^2- bs v^2) has zero bs-radius: ")
cycle2D(1,[u,v],u*u-sign1*v*v,e).radius_sq(es).is_zero()

Cycle (1,u,v,u^2- bs v^2) has zero bs-radius:
Out[3]:
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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