I'm currently working on the iPhone, doing some graphics work for the in-house engine we're developing. At the moment, I'm implementing proper Dot-3 normal mapping - but I've come up against an issue that I haven't been able to solve yet. The drivers for OpenGL ES on the iPhone do not support cube mapping or pixel shaders of any kind, and while I am calculating a normalized vertex position to light position vector for each vertex of a lit mesh, these vectors become unnormalized as they are linearly interpolated across the triangle, making them less than unit length and resulting in much darker lighting than one would expect. Example Considering that the iPhone has only two texture units, and supports only 2D textures, I am trying to figure out a way to create a mapping from an unnormalized 3D vector (the interpolated V->L vector) to a 2D texture coordinate that will give me a correct, normalized representation of the same vector. The technique must be linearly interpolatable, of course! Does anyone have any ideas about how this might be achieved?

Yeah, I'm aware that tessellating the mesh would decrease the noticeability of the issue, but there are likely going to be certain cases where tessellating more may push the renderer over its triangle limits. I'd like to have a solution that works regardless of mesh tessellation.

And as far as I can tell, there's no way to use the results of a previous pass as a texture coordinate, unfortunately...

You could also perform one or two steps of the Newton method.
That is, you try to solve the equation f(t) = dot(tn,tn) - 1 = 0, where t is the scaling factor that normalizes the normal vector n.
Newton's method linearizes this equation: f(t) ~ f(t0) + f'(t0) * (t - t0). t0 is some starting value, in your case probably t0 = 1.
The derivative is f'(t) = 2t*dot(n,n).
With this, you have
f(t0) + f'(t0) * t = t0*t0*dot(n,n) - 1 + 2*t0*dot(n,n) * (t - t0) = 0
--> t = (1 + t0*t0*dot(n,n)) / (2*t0*dot(n,n))
If you plug in t0 = 1, this simplifies to
t = (1 + dot(n,n)) / (2 * dot(n,n))
Compute this t and multiply n by it. Check whether the new "normalized" n gives better results.

Hmmm, can you do that on the iPhone at all? I guess not?

Lutz, it's ALMOST implementable, with a couple passes... The only issue is that there's no way to calculate 1 / (2 * dot(n,n)). No reciprocal functionality with texture combiners that I can see. The rest of it is possible... but kind of useless without the quotient! :(

Make unique normal reScale texture per object, witch are used as scalar for normals. (comensates shrinking normals)

/Tyrian

Ps. normal scalar texture can be pre calculated into dot3 normal map, if texture is unique for object. (just let normals be un-normalized & use bigger value range 0-255 => -5.0 - +5.0)

[Edited by - TyrianFin on June 9, 2009 5:55:55 AM]

The probably cheapest way is to use an object space normal map as TyrianFin suggests. If you can't use that because it's too much texture data or your normal maps are repeating over geometry, you can modify the normalization cubemap approach:

That approach uses a 3d texture to set n = texCube(normalizationSampler, n),
where the normalization texture effectively computes n / sqrt(dot(n,n)).

So instead of a 3d texture, you can use a 1d texture and fill it with values
t(x) = 1 / sqrt(x) for x = 0..1
(in your case, dot(n,n) will always be <= 1 because n is a lerp of two normalized vectors). The pixel shader will then compute
n = n * tex1d(normalizationSampler, float2(dot(n,n),0)).x.
I hope this is implementable on your hardware!

- Lutz

Unfortunately, there's no way to modify the texture coordinate per-pixel. :(