None ex.I.5.28.i

$\sigma$-drawing of certain $\breve{\sigma}$-zero-radius pencils

Exercise I.5.28.i [1] $ \newcommand{\scalar}[3][]{\left\langle #2,#3 \right\rangle_{#1}} \newcommand{\Space}[3][]{\mathbb{#2}^{#3}_{#1}} \newcommand{\Space}[3][]{\mathbb{#2}^{#3}_{#1}} \newcommand{\cycle}[3][]{{#1 C^{#2}_{#3}}} \newcommand{\realline}[3][]{#1 R^{#2}_{#3}} \newcommand{\bs}{\breve{\sigma}} \newcommand{\zcycle}[3][]{#1 Z^{#2}_{#3}} \newcommand{\SL}[1][2]{\mathrm{SL}_{#1}(\Space{R}{})} \newcommand{\rs}{\sigma_r} \newcommand{\lvec}[1]{\overrightarrow{#1}} \newcommand{\rmi}{\mathrm{i}} \newcommand{\alli}{\iota} \newcommand{\rme}{\mathrm{e}} \newcommand{\rmd}{\mathrm{d}} \newcommand{\rmh}{\mathrm{j}} \newcommand{\rmp}{\varepsilon} \newcommand{\modulus}[2][]{\left| #2 \right|_{#1}} \newcommand{\sperp}{⋋} $ A pencil of cycles either contains at most two $\bs$-zero-radius cycles or consists entirely of $\bs$-zero-radius cycles. Moreover, a pencil spanned by two different cycles cannot consist only of e-zero-radius cycles. Describe all pencils consisting only of p- and h-zero-radius cycles.

Hint. Formula (I.5.9) will be useful for describing pencils consisting of (and thus spanned by) $\bs$-zero-radius cycles. Orbits of subgroup $N'$ are an example of pencils of p-zero-radius cycles drawn as parabolas. You can experiment with $\sigma$-drawing of certain $\bs$-zero-radius pencils.

Solution. We will illustrate the statement with drawing from the next routine.

In [1]:
from init_cycle import *
A=asy()
            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 procedure draws some representatives from the linear span
of two cycles it is also used in ex-linear-span.py

In [2]:
def makepencil(C2, C3, xmin, xmax, ymin, ymax, colour1, colour2, colour3,name):
    N=5
    A.erase()
    A.size(100)
    A.send("real u=5mm;")
    A.send(C2.asy_string(xmin, xmax, ymin, ymax, colour1, "1.5pt"))
    A.send(C3.asy_string(xmin, xmax, ymin, ymax, colour2, "1.5pt"))
    for i in range(N):
        C4=cycle2D(C2.exmul(i+1)+C3.exmul(N-i))
        A.send(C4.asy_string(xmin, xmax, ymin, ymax,\
        [colour1[0]*(i+1.0)/N+colour2[0]*(1.0-1.*i/N),\
        colour1[1]*(i+1.0)/N+colour2[1]*(1.0-1.*i/N),\
        colour1[2]*(i+1.0)/N+colour2[2]*(1.0-1.*i/N)]))
        C4=cycle2D(C2.exmul(-pow(i+1.0,1.1)))+C3
        A.send(C4.asy_string(xmin, xmax, ymin, ymax,\
        [colour3[0]*(i+1.0)/N+colour2[0]*(1.0-1.*i/N),\
        colour3[1]*(i+1.0)/N+colour2[1]*(1.0-1.*i/N),\
        colour3[2]*(i+1.0)/N+colour2[2]*(1.0-1.*i/N)]))
        C4=C2+cycle2D(C3.exmul(-pow(i+1.0,1.1)))
        A.send(C4.asy_string(xmin, xmax, ymin, ymax,\
        [colour3[0]*(i+1.0)/N+colour1[0]*(1.0-1.*i/N),\
        colour3[1]*(i+1.0)/N+colour1[1]*(1.0-1.*i/N),\
        colour3[2]*(i+1.0)/N+colour1[2]*(1.0-1.*i/N)]))
    A.shipout(name)
    return(Image(f'{name}.png'))
In [3]:
C2=cycle2D([0,1],e.subs({sign : 0}),0)
C3=cycle2D([0,3],e.subs({sign : 0}),0)

Span which consists of p-zero-radius cycles

In [4]:
makepencil(C2, C3, -3,3,-2,4,[0,1,0],[0,0,1],[1,0,0], "pencilpzr")
Out[4]:
No description has been provided for this image
In [5]:
makepencil(cycle2D(C2,e.subs({sign : -1})), 
           cycle2D(C3,e.subs({sign : -1})), 
           -3,3,-2,4,[0,1,0],[0,0,1],[1,0,0], "pencilpzre")
Out[5]:
No description has been provided for this image
In [6]:
makepencil(cycle2D(C2,e.subs({sign : 1})), 
           cycle2D(C3,e.subs({sign : 1})), 
           -3,3,-2,4,[0,1,0],[0,0,1],[1,0,0], "pencilpzrh")
Out[6]:
No description has been provided for this image
In [7]:
C2=cycle2D([0,0],e.subs({sign : 1}),0)
C3=cycle2D([1,1],e.subs({sign : 1}),0)

Span which consists of h-zero-radius cycles

In [8]:
makepencil(C2, C3, -3,3,-2,4,[0,1,0],[0,0,1],[1,0,0], "pencilhzr")
Out[8]:
No description has been provided for this image
In [9]:
makepencil(cycle2D(C2,e.subs({sign : -1})),
           cycle2D(C3,e.subs({sign : -1})), 
           -3,3,-2,4,[0,1,0],[0,0,1],[1,0,0], "pencilhzre")
Out[9]:
No description has been provided for this image
In [10]:
makepencil(cycle2D(C2,e.subs({sign : 0})), 
           cycle2D(C3,e.subs({sign : 0})), 
           -3,3,-2,4,[0,1,0],[0,0,1],[1,0,0], "pencilhzrp")
Out[10]:
No description has been provided for this image
In [11]:
del(A)
closing Asymptote session

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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