# Interact script based on http://interact.sagemath.org/node/20
#
# Shows the action of a Moebius transformation
#
# / a b \         a*z + b
# |     | * z := ---------,   ad - bc != 0
# \ c d /         c*z + d
#
#
# on the complex plane

def mob(A,z):
    return (A[0][0]*z + A[0][1]) / (A[1][0]*z + A[1][1])

html('<h2> The Action of a Moebius Transformation</h2>')
@interact(layout = [['a','b','c', 'd','r','auto_update']])
def mob_plot(
             a = input_box( 0 + 1*I, label='$a=$', width = 10),
             b = input_box( 0 + 1*I, label='$b=$', width = 10),
             c = input_box(-1 + 0*I, label='$c=$', width = 10),
             d = input_box( 1 + 0*I, label='$d=$', width = 10),
             r = input_box( 4, label='window size', width = 5),
    auto_update=False):
        A = matrix(CC,2,2,[[a,b],[c,d]])
    
        id = complex_plot(lambda z:z,(-r,r),(-r,r) )
        output = complex_plot( lambda z: mob(A,z), (-r,r),(-r,r))
  
        html.table([['The action of $A$:', '$z \mapsto \\frac{%s\cdot z + %s}{%s\cdot z + %s}' % (A[0][0],A[0][1],A[1][0],A[1][1]) ], ['Domain = $\mathbb{C}$','Action = $A(\mathbb{C})$'],[id, output]])