script.py

from pylab import *

def interception(A, B, H, Sa, Sb):
	''' Returns None if A cannot catch B before B reaches H.
		Otherwise it returns the time and position of the interception.
		See attached documentation for justifications. '''
	
	sin_beta = det(array((A-B,H-B))) / ( norm(A-B) * norm(H-B) )
	
	sin_alpha = (Sb / Sa) * sin_beta
	
	if abs(sin_alpha) > 1 :
		
		print "B moves too fast to be ever caught !"
		return None
	
	else:
		
		sin_gamma = ( sin_alpha * sqrt(1 - sin_beta**2)
					+ sin_beta * sqrt(1 - sin_alpha**2) )
		
		BC = norm(B-A) * (sin_alpha / sin_gamma) 
		
		if BC > norm(H-A):
		
			print "B reaches H before being caught by A !"
			return None
		
		else:
			
			t = BC / Sb
			C = B + BC * (H-B)/norm(H-B)
	
	print "A intercepted B !"
	return t,C



##### EXAMPLE

# Define the constants

A = array(( 1.0 , 5.0 ))
B = array(( 4.0 , 1.0 ))
H =  array(( 6.0 , 7.0 ))
Sa = 1.1
Sb = 1.0 

# Find the intersection

t,C = interception(A, B, H, Sa, Sb)

# Plot the results

scatter(*zip(A,B,H,C), s=100, color='r')

for label, point in zip(['A','B','H','C'], [A,B,H,C]):
    annotate( label, xy = point, xytext = (-10, 10),
        textcoords = 'offset points', fontsize = 24)
        
annotate("", xy=H, xytext=B, xycoords='data', textcoords='data',size=20,
         arrowprops=dict(arrowstyle="simple",connectionstyle="arc3"))

annotate("", xy=C, xytext=A, xycoords='data', textcoords='data',size=20,
         arrowprops=dict(arrowstyle="simple",connectionstyle="arc3"))

title("A intercepts B in C", fontsize = 24)

show()