First unification of EPH geometries in 3D and rotations of a cone¶

Exercise I.3.16 [1] Verify that the rotation of a cone's generator corresponds to the Möbius transformations in three planes.

Solution. First we define a procedure to check collinearity of two vectors

from init_cycle import *
def is_collinear(X, Y):
    return bool((X[0]*Y[1]-X[1]*Y[0]).normal().is_zero() \
    and (X[0]*Y[2]-X[2]*Y[0]).normal().is_zero())

            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

It is handy to have precooked $\sqrt{2}$ ready.

sqrt2=pow(2,numeric(1,2))

Calculate components of the Moebius transform with matrix $\begin{pmatrix}a&b\\-b&a\end{pmatrix}$ from $K$

[U,V]=clifford_moebius_map(sl2_clifford([a, b, -b, a], e),[0, v], e)

The cone vertex is $(0, (v+1/v)/2, (v-1/v)/2)$. Vector from the cone's vertix to the elliptic point ($\sigma=-1$).

E=[-U.subs({sign : -1}), (v+1/v)/2-V.subs({sign : -1}), (v-1/v)/2]

Vector from the cone's vertix to the parabolic point ($\sigma=0$).

P=[-U.subs({v : v/sqrt2, sign : 0}), \
(v+1/v)/2-V.subs({v : v/sqrt2, sign : 0})/sqrt2, \
(v-1/v)/2-V.subs({v : v/sqrt2, sign : 0})/sqrt2]

Vector from the cone's vertix to the hyperbolic point ($\sigma==1$).

H=[-U.subs({sign : 1}), (v+1/v)/2, (v-1/v)/2-V.subs({sign : 1})]

Check collinearity of points E and P

"E and P are collinear: %s" % is_collinear(E, P)

'E and P are collinear: True'

Check collinearity E and H

"E and H are collinear: %s" % is_collinear(E, H)

'E and H are collinear: 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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