Question

I’m attempting to implement soft shadows in my raytracer. To do so, I plan to shoot multiple shadow rays from the intersection point towards the area light source. I’m aiming to use a spherical area light–this means I need to generate random points on the sphere for the direction vector of my ray (recall that ray’s are specified with a origin and direction).

I’ve looked around for ways to generate a uniform distribution of random points on a sphere, but they seem a bit more complicated than what I’m looking for. Does anyone know of any methods for generating these points on a sphere? I believe my sphere area light source will simply be defined by its XYZ world coordinates, RGB color value, and r radius.

I was referenced this code from Graphics Gems III, page 126 (which is also the same method discussed here and here):

void random_unit_vector(double v[3]) {    
double theta = random_double(2.0 * PI);
double x = random_double(2.0) - 1.0;
double s = sqrt(1.0 - x * x);
v[0] = x;
v[1] = s * cos(theta);
v[2] = s * sin(theta);

}

This is fine and I understand this, but my sphere light source will be at some point in space specified by 3D X-Y-Z coordinates and a radius. I understand that the formula works for unit spheres, but I’m not sure how the formula accounts for the location of the sphere.

Thanks and I appreciate the help!

Answer 1

You seem to be confusing the formulas that generate a direction — ie., a point on a sphere — and the fact that you’re trying to generate a direction /toward/ a sphere.

The formula you gave samples a random ray uniformly : it finds an X,Y,Z triple on the unit sphere, which can be considered as a direction.

What you actually try to achieve is to still generate a direction (a point on a sphere), but which favors a particular direction that points toward a sphere (or which is restricted to a cone : the cone you obtain from the center of your camera and the silhouette of the sphere light source).
Such thing can be done in two ways :

• Either importance sampling toward the center of your spherical light source with a cosine lobe.
• Uniform sampling in the cone defined above.

In the first cases, the formulas are given in the “Global Illumination COmpendium” :
http://people.cs.kuleuven.be/~philip.dutre/GI/TotalCompendium.pdf
(item 38 page 21)..
In the second case, you could do some rejection sampling, but I’m pretty sure there are some close form formula for that.

Finally, there is a last option : you could use your formula, consider the resulting (X,Y,Z) as a point in your scene, and thus translate it to the position of your sphere, and make a vector pointing from your camera toward it. However, it will pose serious issues :

• You will be generating vectors toward the back of your sphere light
• You won’t have any formula for the pdf of the generated set of directions which you would need for later Monter Carlo integration.