None ex.I.5.18.iv
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We check properties of h-zero-radius cycles

Exercise I.5.18.iv [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}{⋋} $ An h-zero-radius cycle in $\Space[\sigma]{R}{}$ with the e-center $(a,b)$ intersects the real axis at points $(a\pm b,0)$ with slopes $\pm1$. Furthermore, the h-centre of this cycle coincides with its e-focus.

Hint. For slopes, use implicit derivation. Also, for a parabola, it follows from the Exercise I.5.17.iii and the classical reflection property of parabolas.

Solution. Create an h-zero-radius cycle with the e-centre at $(a,b)$

In [1]:
from init_cycle import *
C0=cycle2D([a,b],e,0,es).subs(sign1==1)

            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]:
("h-zero-radius cycle centered at (a,b) intersects the real line at points (a-b) and (a+b): %s" %\
(C0.val([a-b,0]).normal().is_zero() and C0.val([a+b,0]).normal().is_zero()))
Out[2]:
'h-zero-radius cycle centered at (a,b) intersects the real line at points (a-b) and (a+b): True'

Find the implicit derivative

In [3]:
E=C0.val([u,v])
D=E.diff(u)/E.diff(v)
In [4]:
("Slopes of h-zero-radius cycle at the intersection points with the real axis are %s and %s " %\
(D.subs({v : 0, u : a-b}), D.subs({v : 0, u : a+b})))
Out[4]:
'Slopes of h-zero-radius cycle at the intersection points with the real axis are 1 and -1 '
In [5]:
("h-center and e-Focus of this cycle coinside %s %s" % \
(C0.focus(es)[0].is_equal(C0.center()[0]),\
C0.focus(es)[1].subs({sign1 : -1}).is_equal(C0.center()[1].subs({sign : 1}))))
Out[5]:
'h-center and e-Focus of this cycle coinside True 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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