None ex.I.10.13
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Check relation between parabolic Cayley transform and f-orthogonality

Exercise I.10.13 [1] $\newcommand{\Space}[3][]{\mathbb{#2}^{#3}_{#1}} \newcommand{\cycle}[3][]{{#1 C^{#2}_{#3}}} \newcommand{\realline}[3][]{#1 R^{#2}_{#3}} \newcommand{\bs}{\breve{\sigma}}$ An infinitesimal cycle $\cycle{a}{\bs}$ I.7.13 is f-orthogonal (in the sense of Exercise I.7.26.ii) to a cycle $\cycle[\tilde]{a}{\bs}$ if and only if the Cayley transform I.10.9(p) of $\cycle{a}{\bs}$ is f-orthogonal to the Cayley transform of $\cycle[\tilde]{a}{\bs}$.

Solution. We

In [1]:
from init_cycle import *
vp=possymbol("vp")
TC=matrix([[one, -e1], [sign1*e1, one]])

            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

We define the procedure which will do the required steps. Cycle C10 is a cycle of infitesimal radius $\varepsilon$, C11 is the Cayley transform of C10. We output the distance between the focus of C11 and the Cayley transform of focus of C10. The second output gives the condition of f-orthogonality to be of order $O(\epsilon^3)$.

In [2]:
def cayley_ortho(nval):
    epsilon_is_zero = relational(epsilon, 0, eq)
    C10 = cycle2D(1, [u, n],  pow(u,2)-pow(n,2)*sign1+pow(epsilon,2), e).subs(nval)
    C11=cayley(C10,0,sign1)
    display(Latex("Checking infinitesimal cycle: $%s$" % C10.string()))
    displ = (C11.focus(er, True).subs(nval)\
    - clifford_moebius_map(TC, matrix([[u],[vp]]), e)).evalm().normal()
    display(Latex("  Focus of the Cayley-transformed infinitesimal cycle displaced by: ($%s$, $%s$)" %\
    (displ.op(0).subs({sign : 0}, subs_options.algebraic).series(epsilon_is_zero,  2).normal(),\
    displ.op(0).subs({sign : 0}, subs_options.algebraic).series(epsilon_is_zero,  2).normal())))     
    return Latex("  f-orthogonality of Cayley transforms of infinitesimal cycle to other: $%s$" %\
    C11.is_f_orthogonal(cayley(C,0,sign1), es).series(epsilon_is_zero, 3).normal())

Finally we make check in to cases. The first one is for parabolic focus.

In [3]:
cayley_ortho({n : (vp-(pow(pow(vp,2)-pow(epsilon,2)*(sign2-sign1),half)))/(sign2-sign1)})
Checking infinitesimal cycle: $(1, {\left(\begin{array}{cc}u&-\frac{\sqrt{vp^{2}- {(\sigma_2-\sigma_1)} \epsilon^{2}}-vp}{\sigma_2-\sigma_1}\end{array}\right)}^{{symbol114} }, -\frac{ \sigma_1 {(\sqrt{vp^{2}- {(\sigma_2-\sigma_1)} \epsilon^{2}}-vp)}^{2}}{{(\sigma_2-\sigma_1)}^{2}}+u^{2}+\epsilon^{2})$
Focus of the Cayley-transformed infinitesimal cycle displaced by: ($\mathcal{O}(\epsilon^{2})$, $\mathcal{O}(\epsilon^{2})$)
Out[3]:
f-orthogonality of Cayley transforms of infinitesimal cycle to other: ${(0==0)}+{(0==0)} \epsilon+{(\frac{1}{2} (-\frac{2 vp n-m- k u^{2}+2 {l} u}{vp}==0))} \epsilon^{2}+\mathcal{O}(\epsilon^{3})$

The second check is for parabolic focus (i.e. the vertex) of the parabola.

In [4]:
cayley_ortho({n : pow(epsilon,2)/2/vp, sign2 : sign1})
Checking infinitesimal cycle: $(1, {\left(\begin{array}{cc}u&\frac{1}{2} \frac{\epsilon^{2}}{vp}\end{array}\right)}^{{symbol290} }, -\frac{1}{4} \frac{ \epsilon^{4} \sigma_1}{vp^{2}}+u^{2}+\epsilon^{2})$
Focus of the Cayley-transformed infinitesimal cycle displaced by: ($\mathcal{O}(\epsilon^{2})$, $\mathcal{O}(\epsilon^{2})$)
Out[4]:
f-orthogonality of Cayley transforms of infinitesimal cycle to other: ${(0==0)}+{(0==0)} \epsilon+{(\frac{1}{2} (-\frac{2 vp n-m- k u^{2}+2 {l} u}{vp}==0))} \epsilon^{2}+\mathcal{O}(\epsilon^{3})$

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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