Plotting a Möbius transformation

edit

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