"""
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).
"""
from numpy import cos, sin, pi, matrix
import numpy as npy
import numpy.linalg as linalg
from pylab import figure, show
def csqrt(x):
"""
sqrt func that handles returns sqrt(x)j for x<0
"""
if x<0: return complex(0, npy.sqrt(abs(x)))
else: return npy.sqrt(x)
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)
"""
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 + csqrt(tau**2 - 4*delta))/2.
lambda2 = (tau - csqrt(tau**2 - 4*delta))/2.
return lambda1, lambda2
# 2000 random x,y points in the interval[-0.5 ... 0.5]
X1 = matrix(npy.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 * pi/180.
S = matrix([[sx, 0],
[0, sy]])
R = matrix([[cos(theta), -sin(theta)],
[sin(theta), cos(theta)],])
M = 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 = M*X1
# plot the original x,y as green dots and the transformed x, y as red
# dots
fig = figure()
ax = fig.add_subplot(111)
x1 = X1[0].flat
y1 = X1[1].flat
x2 = X2[0].flat
y2 = X2[1].flat
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)
show()
