Relations between tangency and parallelity¶

Exercise I.5.31 [1] Check that:

Straight lines touching the zero-radius cycle at infinity. 2 Two straight lines are parallel if and only if they are touching (at infinity).

Solution. The function to check the tangency, its zero value correspond to tangent cycles

from init_cycle import *
def tangency(C, C1):
    return (pow(C.cycle_product(C1),2)-C.cycle_product(C)*C1.cycle_product(C1))\
    .normal()

            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

Define two straight lines

Cl=cycle(0,[l,n],m,e)
Cl1=cycle(0,[l1,n1],m1,e)

("A cycle is orthogonal to infinity iff it is tangent to infinity: %s"\
% (tangency(C,Zinf)-pow(C.cycle_product(Zinf),2)).normal().is_zero())

'A cycle is orthogonal to infinity iff it is tangent to infinity: True'

Parallel line have equal slops $l/n = l_1/n_1$, or $l n_1-l_1 n=0$.

("Tangency of lines is the same as parallelity: %s" \
% (tangency(Cl,Cl1)-4*sign*pow(l*n1-n*l1,2)).normal().is_zero())

'Tangency of lines is the same as parallelity: 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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