>> y = sqrt(rand(0..1))
>> theta = acos(y)
>> phi = 2*PI*rand(0..1)
>>
>> x = sin(theta)*cos(phi)
>> z = sin(theta)*sin(phi)

> The reason for using sqrt for y is because you want points equally
> distributed over the surface of the sphere, not between y=0 to 1.

Hmm. Ok. I think I got it. So without sqrt() we get too few rays going
through the top part? A few more questions.

1. Is this equally good as the above - at least it looks to be. And a
bit faster.

    double y = sqrt((double) rand()/RAND_MAX);
    double phi = 2*M_PI* (double)rand()/RAND_MAX;
    double x = sqrt(1-y*y)*cos(phi);
    double z = sqrt(1-y*y)*sin(phi);


What is the proper way to actually test the uniformity?

2. I read about "cosine weighted distribution" which should remove the
need to scale with cosine term. This sounds ideal solution. But
according tho this:

http://www.cs.kuleuven.be/~phil/GI/TotalCompendium.pdf

"Generating points uniformly on the disk (see 19), and then projecting
them on the hemisphere, also gives a cosine-weighted distribution of
points on the hemisphere."

I also tried that:


    do{
        x = (double) 2.0*rand()/RAND_MAX -1.0;
        z = (double) 2.0*rand()/RAND_MAX-1.0;
    }
    while(x*x+z*z > 1.0);

    y = sqrt(1.0-(x*x+z*z));

And the results are very close to the 2 other methods.

This is what I read from comp.graphics.algorithms:

"One other way to reduce variance in this situation is to shoot rays
with a cosine-weighted distribution. Generate points distributed
uniformly on the unit disk, then project up to the unit
hemisphere. Each point will define a ray drawn from the proper
distribution. Then, at least for the indirect lighting, you don't need
to include the cosine explicitly at all. But that's not the real win:
the win is that each sample actually contains more information about
the mean value, so variance will be less with the same number of rays."

So what is the truth here? Do I need to account the angle of the ray or
not?

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