#! /usr/bin/env python3
#
import matplotlib.pyplot as plt
import numpy as np
#
print ( 'simulate_compare: compare simulation histogram to exact pdf' )
#
#  n = how many samples we want to create.
#
n = 10000
#
#  p1 and p2 record the random locations of the two flies.
#  We do this n times.
#
p1 = np.random.rand ( 2, n )
p2 = np.random.rand ( 2, n )
#
#  d records how far apart the flies are.
#
d = np.linalg.norm ( p1 - p2, axis = 0 )
#
#  Now we plot a histogram of the values of d.
#
plt.hist ( d, bins = 20, rwidth = 0.95, density = True )
#
#  Now we compute 101 values of the mathematical function (PDF)
#  that the histogram approximates.
#
n = 101
d = np.linspace ( 0.0, np.sqrt ( 2.0 ), n )
pdf = np.zeros ( n )
i1 = np.where ( d <= 1.0 )
pdf[i1] = 2 * d[i1] * ( d[i1]**2 - 4 * d[i1] + np.pi )
i2 = np.where ( 1.0 < d )
pdf[i2] = 2 * d[i2] * (4 * np.sqrt ( d[i2]**2 - 1 ) - ( d[i2]**2 + 2 - np.pi ) \
    - 4.0 * np.arctan ( np.sqrt ( d[i2]**2 - 1 ) ) )
#
#  Plot the PDF.
#
plt.plot ( d, pdf, 'r-', linewidth = 3 )

plt.grid ( True )
plt.xlabel ( '<-- Distance -->' )
plt.ylabel ( '<-- Frequency -->' )
plt.title ( 'Compare exact and observed PDF' )
filename = 'simulate_compare.png'
plt.savefig ( filename )
print ( '  Graphics saved as "%s"' % ( filename ) )
plt.show ( )