None ex.I.10.12
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The family of infinitesimal cycles is invariant under the Cayley transform

Exercise I.10.12. [1] Show that images of infinitesimal cycles under the parabolic Cayley transform are, themselves, infinitesimal cycles.

Solution. First we define some suitable substitutions to simplify algebraic equations later.

In [1]:
from init_cycle import *
sign_cube = {pow(sign, 3)  : sign}
sign1_cube = {pow(sign1, 3)  : sign1}
vp=possymbol("vp")
sl2_relation = {c*b  : a*d-1}
sl2_relation1 = {a  : (1+b*c)/d}

            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

The check will be performed by the next procedure. It verifies the expression of the Cayley transform on infinitesimal cycles and evaluates its radius.

In [2]:
def infinitesimal_trans(nval):
    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()))
    display(Latex("  Image under Cayley transform of infinitesimal cycle has radius squared: $%s$" %\
    C11.radius_sq(es).subs(sl2_relation1, subs_options.algebraic).\
    subs(sign_cube, subs_options.algebraic).series(relational(epsilon, 0, eq), 3).normal()))

Now we do first check, the first for elliptic focus.

In [3]:
infinitesimal_trans({n : (vp-(pow(pow(vp,2)-pow(epsilon,2)*(sign2-sign1),half)))/(sign2-sign1)})
Checking infinitesimal cycle: $(1, {\left(\begin{array}{cc}u&-\frac{vp-\sqrt{ {(\sigma_1-\sigma_2)} \epsilon^{2}+vp^{2}}}{\sigma_1-\sigma_2}\end{array}\right)}^{{symbol114} }, u^{2}+\epsilon^{2}-\frac{ \sigma_1 {(vp-\sqrt{ {(\sigma_1-\sigma_2)} \epsilon^{2}+vp^{2}})}^{2}}{{(\sigma_1-\sigma_2)}^{2}})$
Image under Cayley transform of infinitesimal cycle has radius squared: ${(-\frac{-1+vp- u^{2} \sigma_1}{vp})} \epsilon^{2}+\mathcal{O}(\epsilon^{3})$

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

In [4]:
infinitesimal_trans({n : pow(epsilon,2)/2/vp, sign2 : sign1})
Checking infinitesimal cycle: $(1, {\left(\begin{array}{cc}u&\frac{1}{2} \frac{\epsilon^{2}}{vp}\end{array}\right)}^{{symbol212} }, u^{2}+\epsilon^{2}-\frac{1}{4} \frac{ \sigma_1 \epsilon^{4}}{vp^{2}})$
Image under Cayley transform of infinitesimal cycle has radius squared: ${(-\frac{-1+vp- u^{2} \sigma_1}{vp})} \epsilon^{2}+\mathcal{O}(\epsilon^{3})$

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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