Check the basic properties of geodesics¶

Exercise I.9.20 [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}} $ Check that the main parabolas passing the parabolic imaginary unit $\rmp$ \begin{equation} \label{eq:principle} \rs u^{2}-4v+4=0, \end{equation} where $\rs=-1$, $0$, $1$, are f-orthogonal to the real line. Their rotations by an element of $ N'$ are parabolas.

Solution. Check f-orthogonality condition:

from init_cycle import *
C0=cycle2D(sign1,[0,n],n*n,e)
print("Geodesic is f-orthogonal to the real line: %s" % bool(C0.is_f_orthogonal(real_line,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
Geodesic is f-orthogonal to the real line: True

We are producing the rotated cycles:

Latex("N'-rotated geodesics are: $%s$ " %C0.matrix_similarity(sl2_clifford(1,0,t,1,e),e).subs(n==2).string())

This notebook is a part of the MoebInv notebooks project [2] .

References¶

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.
Vladimir V. Kisil, MoebInv notebooks, 2019.

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