While taking an excellent math course over at fxPhd someone asked about the math behind Maya’s sphRand command, and it kicked off a back and forth exploration into the difficulties behind getting an even distribution of random points on a sphere. This is useful for any number of applications where random vectors are needed.
[Note: This topic has been covered (better) elsewhere on the web, but I thought it was interesting enough to preserve for myself on this blog. -am]
So how do you generate a random vector in 3D space? The natural inclination is to generate each point based on a simple random azimuth and elevation. But because the circumference of latitudes decreases from the equator towards the poles, the result is an uneven distribution. Generate a few vectors and things will seem fine. But if you generate a large set of thousands of points, you get this:
But we need an even distribution. How about this: generate random values between -1 and 1 for each X, Y and Z. Then normalize those vectors, and again you have a collection of random points on a sphere. But the distribution still isn’t even… you are basically projecting the points of a cubic volume onto the sphere surface:
You can solve this problem by culling points outside the radius of the sphere (magnitude >1) as they are being generated, and the result is an even distribution on a sphere. This is known as the rejection method, and it works, but it’s inefficient. You are computing a lot of rejected points. Additionally, as a practical matter “N iterations” yields some number less than “N” vectors, making it difficult to specify the total number of vectors to be created by such a function.
Enter the trigonometric method (something of a misnomer seeing as how the method involves the integral of the surface of revolution.)
θ = υ(-1,1)
And Φ = υ(0,2π)
Where υ(a,b) is equal to a uniform random scalar.
So, go from Polar to Cartesian coordinates, and you get:
x = cos(θ) cos(Φ)
y = sin(Φ)
z = cos(θ) sin(Φ)
And it works.
Digression: if you want to get down to the heart of things, getting a random scalar value, or more specifically a repeatable pseudo-random value based on a specified seed, is it’s own challenge. Here’s a snippet of GLSL code, a function written by Stephen Gustavson which generates a “random” scalar value from a scalar seed input… (cos and mod should be familiar…the “fract” function returns the fractional (decimal) portion of it’s input.)
float randomizer(const float x)
float z = mod(x, 5612.0);
z = mod(z, 3.1415927 * 2.0);
return(fract(cos(z) * 56812.5453));