Check properties of reflection with respect to orthogonality¶

Exercise I.6.26 [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}} $ Let $\det \cycle[\tilde]{s}{\bs}\neq 0$. Then, for cycle similarity with $\cycle[\tilde]{s}{\bs}$:

Verify the identities $-\cycle[\tilde]{s_1}{\bs} \overline {\cycle[\tilde]{s}{\bs}}\cycle[\tilde]{s_1}{\bs} ={ \det_{\bs}}(\cycle[\tilde]{s_1}{\bs})\cdot \cycle[\tilde]{s_1}{\bs} $ and $ -\cycle[\tilde]{}{\bs}\overline {\cycle{}{\bs}}\cycle[\tilde]{}{\bs} =-{\det_{\bs}}(\cycle[\tilde]{}{\bs})\cdot\cycle{}{\bs}, $ where $\cycle{s}{\bs}$ is a cycle $\bs$-orthogonal to $\strut \cycle[\tilde]{}{\bs}$. Note the difference in the signs in the right-hand sides of both identities.
Describe all cycles which are fixed (as points in the projective space $\Space{P}{3}$) by the similarity with the given cycle $\cycle[\tilde]{s}{\bs}$.

Hint. Use a decomposition of a generic cycle into a sum $\cycle[\tilde]{s}{\bs}$ and a cycle orthogonal to $\cycle[\tilde]{s}{\bs}$.

Solution. The direct check:

from init_cycle import *
print("C conjugated with itself is C multiplied by det(C): %s" % \
C.cycle_similarity(C,e,sign_mat1,sign_mat1,pow(sign_mat1,-1).evalm())\
.subs(pow(s1,2)==1,subs_options.algebraic)\
.is_equal(C.exmul(C.hdet(e,sign_mat1)).subs(pow(s1,2)==1,subs_options.algebraic),False))

            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
C conjugated with itself is C multiplied by det(C): True

A similar check for orthogonal cycles

Cn=C1.subject_to(C1.is_orthogonal(C,e))
print("A cycle orthogonal to C conjugated with C is itself times -det(C): %s " % \
Cn.cycle_similarity(C,e).is_equal(Cn.exmul(-C.hdet(e)),False))

A cycle orthogonal to C conjugated with C is itself times -det(C): True

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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