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Plotting a Möbius transformation

asked 2023-11-27 15:08:35 +0100

toni gravatar image

updated 2024-02-12 11:35:59 +0100

FrédéricC gravatar image

This is the equation in question, although I can see it coming that instead some complex exponential form is going to crop up...

Ideally, I would like to reproduce something like this, but a more realistic start could be simply the use of a complex plot along the lines of complex_plot((3 * x + 7) / (-3 * x - 9), (-5,5), (0,5)). Evidently this is not a good example - for instance, I would like to know how to restrict the domain to the positive imaginary components.

Is there a well-known way of handling these transformations?

Here is on possible solution.

(I'm working on understanding the basics of modular forms).

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answered 2023-11-29 15:19:40 +0100

achrzesz gravatar image

updated 2023-11-29 16:26:36 +0100

This version looks nicer:

complex_plot((3 * x + 7) / (-3 * x - 9), (-4,-1.5), (-1,1),
    contoured=True,plot_points=300,tiled=True)

Here is the image of two rectangles

M = Manifold(2, 'M')                  
X.<x,y> = M.chart()                
N = Manifold(2,'N')
U.<u,v> = N.chart()
z=(x+I*y)
def f(z):
    return (3*z+7)/(-3*z-9)
F = M.diff_map(N,[f(z).real(),f(z).imag()], name='F')
p1=X.plot(U,mapping=F,number_values={x:40,y:40},
      ranges={x:(-2.7,2.7),y:(-2.7,2.7)},
      color='grey',thickness=1,label_axes=False) 
p2=X.plot(U,mapping=F,number_values={x:40,y:40},
      ranges={x:(-8.7,-3.3),y:(-2.5,2.5)}, 
      color='grey',thickness=1,label_axes=False) 
p1+p2
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Asked: 2023-11-27 15:08:35 +0100

Seen: 163 times

Last updated: Nov 29 '23