Images of the rectangular grid under the inversion¶

Exercise I.6.25 [1] Check that the rectangular grid is produced by horizontal and vertical lines given by quadruples $(0,0,1,m)$ and $(0,1,0,m)$, respectively.

The similarity with the cycle $(1,0,0,-1)$ sends a cycle $(k,l,n,m)$ to $(m,l,n,k)$. In particular, the image of the grid are cycles $(m,0,1,0)$ and $(m,1,0,0)$.

Solution. Define horizontal and vertical lines

from init_cycle import *
Ch=cycle2D(0,[0,1],m,e)
Cv=cycle2D(0,[1,0],m1,e)

            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

Check their orthogonality

print("Veritcal and horizontal lines are orthogonal in any case: %s" % bool(Ch.is_orthogonal(Cv)))

Veritcal and horizontal lines are orthogonal in any case: True

Elliptic and hyperbolic unit cycle

Cu=cycle2D(1,[0,0],-1,e)

For curiosity we may want to see their equations.

display(Latex("Image of horizontal lines (EH cases): $%s$" % Ch.cycle_similarity(Cu).string()))
Latex("Image of vertical lines (EH cases): $%s$" % Cv.cycle_similarity(Cu).string())

Latex("Image of generic cycle under inversion: $%s$" % C.cycle_similarity(Cu).string())

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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