A friend was writing a computer game in his spare time and wanted to speckle a sphere with texture uniformly over its surface. Idly, he asked our team how to do this. We came up with some naive ideas like taking the x,y and z co-ordinates from a uniform sample and normalizing them to make a vector or length r, the radius of the sphere. In Python, this would be:

import random, math, pylab, mpl_toolkits.mplot3d

x_list, y_list, z_list = [],[],[]

nsamples = 10000

for sample in xrange(nsamples):

x, y, z = random.uniform(-1.0, 1.0), random.uniform(-1.0, 1.0), random.uniform(-1.0, 1.0)

radius = math.sqrt(x ** 2 + y ** 2 + z ** 2)

x_list.append(x / radius)

y_list.append(y / radius)

z_list.append(z / radius)

fig = pylab.figure()

ax = fig.gca(projection='3d')

ax.set_aspect('equal')

pylab.plot(x_list, y_list, z_list, '+')

pylab.show()

but this is not quite uniformly distributed over the sphere. It produces a sphere like this:

Sphere with Cartesian co-ordinates taken from a uniform sample and then normalized |

Using polar co-ordinates and uniformly sampling over 2π doesn't make things better either. This:

for sample in xrange(nsamples):

phi, theta = random.uniform(0, 2.0) * math.pi, random.uniform(0, 2.0) * math.pi

x_list.append(math.cos(phi) * math.cos(theta))

y_list.append(math.cos(phi) * math.sin(theta))

z_list.append(math.sin(phi))

gives:

Sphere with polar co-ordinates taken from a uniform sample |

The solution involves Gaussian (a.k.a 'Normal') distributions, thus:

for sample in xrange(nsamples):

x, y, z = random.gauss(0.0, 1.0), random.gauss(0.0, 1.0), random.gauss(0.0, 1.0)

radius = math.sqrt(x ** 2 + y ** 2 + z ** 2)

x_list.append(x / radius)

y_list.append(y / radius)

z_list.append(z / radius)

*Why*this works I'll leave to another post. Suffice to say this even distribution is used not just in games but in thermodynamics where you might want to model the velocities of molecules of a gas. In this scenario, (x ** 2 + y ** 2 + z ** 2) would model the squared velocity of a molecule, useful in calculating the energy, ½mv

^{2}

## No comments:

## Post a Comment