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Check properties of the zero-radius cycle at infinity

Exercise I.8.2 [1] Check the following:

  1. $Z_{\infty}$ is the image of the zero-radius cycle $Z_{(0,0)}=(1,0,0,0)$ at the origin under reflection into the unit cycle $(1, 0,0,-1)$---see blue cycles in Fig.~I.6.4(b)--(d) from [1] .

  2. The following statements are equivalent:

    • A point $(u,v)\in\mathbb{R}_\sigma$ belongs to the \emph{zero-radius cycle} $Z_{(0,0)}$ centred at the origin.
    • The zero-radius cycle $Z_{(u,v)}$ is $\sigma$-orthogonal to the zero-radius cycle $Z_{(0,0)}$.
    • The inversion $z\mapsto \frac{1}{z}$ in the unit cycle is singular in the point $(u,v)$.
    • The image of $Z_{(u,v)}$ under inversion in the unit cycle is orthogonal to $Z_{\infty}$.

    If any one of the above statements is true, we also say that the image of $(u,v)$ under inversion in the unit cycle belongs to the zero-radius cycle at infinity.

Hint. These can be easily obtained by direct calculations, even without a CAS.

Solution. We define the required cycles:

In [1]:
from init_cycle import *
Z0=cycle2D([0,0],e,0)
Cu=cycle2D(1,[0,0],-1,e)

            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

Then we simply output the reuired statements:

In [2]:
Latex("Zero cycle at infinity is the reflection of zero cycle at origin: %s" % \
Zinf.is_equal(Z0.cycle_similarity(Cu)))
Out[2]:
Zero cycle at infinity is the reflection of zero cycle at origin: True
In [3]:
Latex(f"$(u,v)$ belongs to zero cycle at the origin if: ${Z0.val([u,v])==0}$")
Out[3]:
$(u,v)$ belongs to zero cycle at the origin if: $u^{2}- \sigma v^{2}==0$
In [4]:
Latex(f"Zero cycle at $(u,v)$ is orthogonal to zero cycle at the origin if: ${Z.is_orthogonal(Z0,es)}$")
Out[4]:
Zero cycle at $(u,v)$ is orthogonal to zero cycle at the origin if: $u^{2}- \sigma v^{2}==0$
In [5]:
Latex(f"Zero cycle at infinity is orthogonal to reflection of $Z(u,v)$ if: ${Zinf.is_orthogonal(Z.cycle_similarity(Cu))}$")
Out[5]:
Zero cycle at infinity is orthogonal to reflection of $Z(u,v)$ if: $u^{2}- \sigma v^{2}==0$

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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