from piscript.PiModule import *
import math

init("cornu.eps", 100, 100)

def fresnel(x):
	return [math.cos(x*x), math.sin(x*x)]

beginpage()
center()
scale(42)

gsave()
newpath()
for i in range(-1,2):
	moveto(-1.25, i)
	lineto(1.25, i)
	moveto(i, -1.25) 
	lineto(i, 1.25) 
stroke(0.8)
grestore()

"""
	plotting a path integral F(T) = \int_{0}^{T} f(t) dt
	A perfect setup for Bezier, since F' = f is given in advance
	DF =  [ f(t) + 4 f(t+dt/2) + f(t+dt)] [ dt / 6 ]
	i.e. uses Simpson's rule
	there is a square root of pi built into
	the Bezier construction
"""

# goes through N cycles of M segments each
N=20
M=4

for k in range(2):

	gsave()
	# X Y = position, x y = speed
	X = 0
	Y = 0
	g = 1
	setgray(g)
	q = 0.97 
	for i in range(N*M):
		t = math.sqrt((2*math.pi*i)/M)
		nt = math.sqrt((2*math.pi*(i+1))/M)
		f = fresnel(t)
		x = f[0]; y = f[1]
		dt = nt-t
		# now build one Bezier segment
		newpath()
		moveto(X, Y)
		X1 = X + x*dt/3.0
		Y1 = Y + y*dt/3.0
		f1 = fresnel(t+0.5*dt)
		x1 = f1[0]; y1 = f1[1]
		t += dt
		f2 = fresnel(t)
		x2 = f2[0]; y2 = f2[1]
		X += (x + 4*x1 + x2)*dt/6.0
		Y += (y + 4*y1 + y2)*dt/6.0
		X2 = X - x2*dt/3.0
		Y2 = Y - y2*dt/3.0
  		curveto(X1, Y1, X2, Y2, X, Y)
		stroke(1-g)
		g *= q
		scalelinewidth(0.9)
	grestore()
	rotate(math.pi)

translate(-0.92, 0.18)
setcolor(0.8, 0, 0)
scale(0.025)
place(texinsert("Cornu spiral"))

endpage()
flush()