"""
Moire patterns from random dot fields
http://en.wikipedia.org/wiki/Moir%C3%A9_pattern
See L. Glass. 'Moire effect from random dots' Nature 223, 578580 (1969).
"""
import cmath
import numpy as np
import numpy.linalg as linalg
import matplotlib.pyplot as plt
def myeig(M):
"""
compute eigen values and eigenvectors analytically
Solve quadratic:
lamba^2 - tau*lambda +/- Delta = 0
where tau = trace(M) and Delta = Determinant(M)
if M = | a b |
| c d |
the trace is a+d and the determinant is a*d-b*c
Return value is lambda1, lambda2
"""
a,b = M[0,0], M[0,1]
c,d = M[1,0], M[1,1]
tau = a+d # the trace
delta = a*d-b*c # the determinant
lambda1 = (tau + cmath.sqrt(tau**2 - 4*delta))/2.
lambda2 = (tau - cmath.sqrt(tau**2 - 4*delta))/2.
return lambda1, lambda2
# 2000 random x,y points in the interval[-0.5 ... 0.5]
X1 = np.random.rand(2,2000)-0.5
#name = 'saddle'
#sx, sy, angle = 1.05, 0.95, 0.
name = 'center'
sx, sy, angle = 1., 1., 2.5
#name= 'stable focus' # spiral
#sx, sy, angle = 0.95, 0.95, 2.5
theta = angle * cmath.pi/180.
S = np.array([[sx, 0],
[0, sy]])
R = np.array([[np.cos(theta), -np.sin(theta)],
[np.sin(theta), np.cos(theta)],])
M = np.dot(S, R) # rotate then stretch
# compute the eigenvalues using numpy linear algebra
vals, vecs = linalg.eig(M)
print 'numpy eigenvalues', vals
# compare with the analytic values from myeig
avals = myeig(M)
print 'analytic eigenvalues', avals
# transform X1 by the matrix
X2 = np.dot(M, X1)
# plot the original x,y as green dots and the transformed x, y as red
# dots
fig = plt.figure()
ax = fig.add_subplot(111)
x1 = X1[0]
y1 = X1[1]
x2 = X2[0]
y2 = X2[1]
ax = fig.add_subplot(111)
line1, line2 = ax.plot(x1, y1, 'go', x2, y2, 'ro', markersize=2)
ax.set_title(name)
fig.savefig('glass_dots1.png', dpi=100)
fig.savefig('glass_dots1.eps', dpi=100)
plt.show()
