# Dynamical Billiards Simulation Map (Fractal?)
# FB - 201011066
# More info:
# http://en.wikipedia.org/wiki/Dynamical_billiards
# http://www.scholarpedia.org/article/Dynamical_billiards
import math
import random
import time
from PIL import Image, ImageDraw
imgx = 300
imgy = 200
image = Image.new("RGB", (imgx, imgy))
draw = ImageDraw.Draw(image)
maxSteps = 200 # of steps of ball motion (in constant speed)
n = random.randint(1, 7) # of circular obstacles
crMax = int(min(imgx - 1, imgy - 1) / 4) # max circle radius
crMin = 10 # min circle radius
# create circular obstacle(s)
cxList = []
cyList = []
crList = []
for i in range(n):
while(True): # circle(s) must not overlap
cr = random.randint(crMin, crMax) # circle radius
cx = random.randint(cr, imgx - 1 - cr) # circle center x
cy = random.randint(cr, imgy - 1 - cr) # circle center y
flag = True
if i > 0:
for j in range(i):
if math.hypot(cx - cxList[j], cy - cyList[j]) < cr + crList[j]:
flag = False
break
if flag == True:
break
draw.ellipse((cx - cr, cy - cr, cx + cr, cy + cr))
cxList.append(cx)
cyList.append(cy)
crList.append(cr)
# initial direction of the ball
a = 2.0 * math.pi * random.random()
t0 = time.time()
for y0 in range(imgy):
for x0 in range(imgx):
x = float(x0)
y = float(y0)
s = math.sin(a)
c = math.cos(a)
# print '%completed' every 10 seconds
t = time.time()
if t - t0 >= 10:
print '%' + str(int(100 * (imgx * y0 + x0) / (imgx * imgy)))
t0 = t
# initial location of the ball must be outside of the circle(s)
flag = True
for i in range(n):
if math.hypot(x - cxList[i], y - cyList[i]) <= crList[i]:
flag = False
break
if flag:
for i in range(maxSteps):
xnew = x + c
ynew = y + s
# reflection from the walls
if xnew < 0 or xnew > imgx - 1:
c = -c
xnew = x
if ynew < 0 or ynew > imgy - 1:
s = -s
ynew = y
# reflection from the circle(s)
for i in range(n):
if math.hypot(xnew - cxList[i], ynew - cyList[i]) <= crList[i]:
# angle of the circle point
ca = math.atan2(ynew - cyList[i], xnew - cxList[i])
# reversed collision angle of the ball
rca = math.atan2(-s, -c)
# reflection angle of the ball
rab = rca + (ca - rca) * 2
s = math.sin(rab)
c = math.cos(rab)
xnew = x
ynew = y
x = xnew
y = ynew
# Plot the starting point according to the final point!
# The color can be decided in many different ways:
# absolute distance method
d = math.hypot(x, y) / math.hypot(imgx - 1, imgy - 1)
# relative distance method
# d = math.hypot(x - x0, y - y0) / math.hypot(imgx - 1, imgy - 1)
# absolute angle method by taking the image center as origin
# ang = math.atan2(x - (imgx - 1) / 2, y - (imgy - 1) / 2)
# relative angle method by taking the starting point as origin
# ang = math.atan2(x - x0, y - y0)
# convert the angle from -pi..pi to 0..2pi
# if ang < 0: ang = 2 * math.pi - math.fabs(ang)
# d = ang / 2 * math.pi # convert the angle to 0..1
k = int(d * 255)
rd = k % 8 * 32
gr = k % 16 * 16
bl = k % 32 * 16
image.putpixel((x0, y0), (rd, gr, bl))
print 'calculations completed'
image.save("Dynamical_Billiards_Map.png", "PNG")
Diff to Previous Revision
--- revision 1 2010-11-06 07:06:23
+++ revision 2 2010-11-06 07:17:11
@@ -42,8 +42,6 @@
# initial direction of the ball
a = 2.0 * math.pi * random.random()
-s = math.sin(a)
-c = math.cos(a)
t0 = time.time()
@@ -51,6 +49,8 @@
for x0 in range(imgx):
x = float(x0)
y = float(y0)
+ s = math.sin(a)
+ c = math.cos(a)
# print '%completed' every 10 seconds
t = time.time()
@@ -102,12 +102,12 @@
# relative distance method
# d = math.hypot(x - x0, y - y0) / math.hypot(imgx - 1, imgy - 1)
# absolute angle method by taking the image center as origin
- # a = math.atan2(x - (imgx - 1) / 2, y - (imgy - 1) / 2)
+ # ang = math.atan2(x - (imgx - 1) / 2, y - (imgy - 1) / 2)
# relative angle method by taking the starting point as origin
- # a = math.atan2(x - x0, y - y0)
+ # ang = math.atan2(x - x0, y - y0)
# convert the angle from -pi..pi to 0..2pi
- # if a < 0: a = 2 * math.pi - math.fabs(a)
- # d = a / 2 * math.pi # convert the angle to 0..1
+ # if ang < 0: ang = 2 * math.pi - math.fabs(ang)
+ # d = ang / 2 * math.pi # convert the angle to 0..1
k = int(d * 255)
rd = k % 8 * 32