## Graphics Trick: Normals from Height

I’ve seen code floating around that computes a normal map from a height map by estimating tangent vectors and doing a cross product. That includes a bit of unnecessary extra math, so here’s a better way to compute the same thing.

First, let’s write the height field as a function of x & y (ok, it’s a texture, so call them u and v if it makes you feel better). So a height field is basically a texture that defines

z = h(x,y)

The tangent-based code already estimated the partial derivatives of h(x,y) with respect to x and y. Let’s call those h_{x} and h_{y}. There are several ways to do this: forward differences

h_{x} = h(x+1) – h(x)

central differences:

h_{x} = (h(x+1) – h(x-1)) / 2

Sobel filter (no, I’m not going to write that one out), or other derivative of a filter.

The trick comes in rewriting this thing as an implicit function of x,y,z that’s 0 on the surface, negative under the surface and positive outside the surface. That’s actually pretty easy:

f(x,y,z) = z – h(x,y)

So the neat thing is that the normal to an implicit surface like this is just the gradient of the implicit function:

N = (f_{x}, f_{y}, f_{z}) = (-h_{x}, -h_{y}, 1)

That’s it. Much friendlier than cross( (1,0,h_{x}), (0,1,h_{y}) ). You could actually get exactly the same result by expanding out the cross product, but I think it’s much cooler (and more applicable in other situations) to know about the normal to an implicit function trick.

Marc