Explicit expression for cycle product¶

Exercise I.5.12.ii [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}} $ The $\bs$-cycle product of cycles defined by quadruples $(k,l,n,m)$ and $(\tilde{k},\tilde{l},\tilde{n},\tilde{m})$ is given by \begin{eqnarray} \label{eq:cycle-product-expl} -2\tilde{l}l+2\bs s^2\tilde{n}n+\tilde{k}m+\tilde{m}k. \end{eqnarray}

Solution. This is one-line calculation, of course:

from init_cycle import *
Latex("Cycle product is: $%s$" % C.cycle_product(C1,es))

            Python wrappers for MoibInv Library
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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.
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Using vector formalism and idx

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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