Graphics Trick: Homogeneous fun

Anyone who’s done any 3D graphics knows how to use homogeneous coordinates for translation and perspective, but they’re good for other places where you’d rather put off a division or avoid it entirely.

Basic homogeneous coordinates use four coordinates (I’ll use lower case to keep things straight) to represent a 3D point (here in upper case)

p = [x y z w]
P = [X  Y  Z] = [x/w  y/w  z/w]

Or more compactly, in shader notation

P = p.xyz / p.w

For regular points, we use w=1. If w=1/2, we get a point twice as far out in the p.xyz direction. If it’s 1/3, we get a point three times as far out in the p.xyz direction. In the limit, as w goes to 0, we get a point infinitely far away in the p.xyz direction.There’s no need to worry about a pesky division by 0, as long as we don’t ever actually have to do the division for the evil w’s.

That’s the basic idea behind my homogeneous rasterization paper (sometimes called clipless rasterization), used in quite a few GPUs. You normally compute the intersection between each triangle edge and a clipping plane to throw away the parts behind you where w might be 0 or negative. If you put off the division and keep everything in homogeneous coordinates long enough, eventually you can get all the way to visible pixels on the screen, with w’s guaranteed to be positive, before you actually have to divide.

But that’s not what this graphics trick is about. What else can we do using homogeneous coordinates to avoid division? Let’s say we want to subtract two points

V = Q – P = q.xyz/q.w – p.xyz/p.w

I can avoid dividing by putting that over a common denominator

V = (q.xyz * p.w  –  p.xyz * q.w) / (q.w*p.w)
v.xyz = q.xyz*p.w – p.xyz*q.w;  v.w = q.w*p.w

That lets either p or q be infinitely far away. So if P is a point on a surface, and Q is the location of a light, then V is a vector from the surface to the light (I’d use the traditional L, but lower-case l and number 1 tend to look a bit too similar). This will handle both point and directional lights with the same code. Of course, if you know a particular w is always 1 or always 0, you’d factor that constant into the equation in your code. Assuming p.w=1 to make things a little easier, for point lights (q.w=1), that reduces to a vector subtraction between the surface and light location

v.xyz = q.xyz – p.xyz;  v.w = 1

For directional lights, which are infinitely far away in a given direction (often used to approximate the sun), q.w=0, and we see the exact surface position doesn’t matter

v.xyz = q.xyz;  v.w=0

Also, though less common, you can use the same methods do parallel and perspective projection with the same code.

So that puts off the division and keeps everything in homogeneous coordinates, but in many lighting computations, you want a unit length vector pointing in the given direction

normalize(V) = V/length(V)

When you normalize, any scale factor that affects all three components has no effect on the final direction. If V is twice as long, or three times as long, or infinitely long, it still points the same way. In particular, as long as we avoid negative w’s, the division by v.w doesn’t affect the final direction, even if v.w=0.

normalize(v.xyz/v.w) = normalize(v.xyz) = normalize(q.xyz * p.w  –  p.xyz * q.w)

It’s amazing how far you can get with two simple rules: if the division might blow up, put it off. If a constant scaling won’t affect the result, don’t do it.

Marc