Box-Muller Transformation

The Box-Muller transform is a method for generating normally distributed random numbers from uniformly distributed random numbers. The Box-Muller transformation can be summarized as follows, suppose u₁ and u₂ are independent random variables that are uniformly distributed between 0 and 1 and let

then z₁ and z₂ are independent random variables with a standard normal distribution. Intuitively, the transformation maps each circle of points around the origin to another circle of points around the origin where larger outer circles are mapped to closely-spaced inner circles and inner circles to outer circles.
Let's see a Python snippet that implements the transformation:

from numpy import random, sqrt, log, sin, cos, pi
from pylab import show,hist,subplot,figure

# transformation function
def gaussian(u1,u2):
  z1 = sqrt(-2*log(u1))*cos(2*pi*u2)
  z2 = sqrt(-2*log(u1))*sin(2*pi*u2)
  return z1,z2

# uniformly distributed values between 0 and 1
u1 = random.rand(1000)
u2 = random.rand(1000)

# run the transformation
z1,z2 = gaussian(u1,u2)

# plotting the values before and after the transformation
figure()
subplot(221) # the first row of graphs
hist(u1)     # contains the histograms of u1 and u2 
subplot(222)
hist(u2)
subplot(223) # the second contains
hist(z1)     # the histograms of z1 and z2
subplot(224)
hist(z2)
show()

The result should be similar to the following:

In the first row of the graph we can see, respectively, the histograms of u₁ and u₂ before the transformation and in the second row we can see the values after the transformation, respectively z₁ and z₂. We can observe that the values before the transformation are distributed uniformly while the histograms of the values after the transformation have the typical Gaussian shape.

Jukowsi Transformation

The following code applies the Jukowsi transformation to various circles on the complex plane.

from pylab import plot,show,axis,subplot
import cmath
from numpy import *

def cplxcircm(radius,center):
 """Returns an array z where every z[i] is a point 
    on the circle's circumference on the complex plane"""
 teta = linspace(0,pi*2,150)
 z = []
 for a in teta:
  # rotating of radius+0j of a radians and translation
  zc = (radius+0j)*cmath.exp(complex(0,a)) - center
  z.append(zc)
 return array( z, dtype=complex )

def jukowsi(z,L):
 return z+L/z # asimmetric Jukowski transformation

# let's try the transformation on three different circles
subplot(131)
z = cplxcircm(1,complex(0.5,0)) #1st circle
w = jukowsi(z,1) # applying the trasformation
plot(real(z),imag(z),'b',real(w),imag(w),'r')
axis('equal')

subplot(132)
z = cplxcircm(0.8,complex(0.4,0.3))
w = jukowsi(z,0.188) 
plot(real(z),imag(z),'b',real(w),imag(w),'r')
axis('equal')

subplot(133)
z = cplxcircm(1.1,complex(0.1,0.0))
w = jukowsi(z,1)
plot(real(z),imag(z),'b',real(w),imag(w),'r')
axis('equal')
show()

The script shows the following figure. Every subplot shows two curves on the complex plane (x real values, y imaginary values), the blue curve is the starting curve and the red is the transformation.