I did notice however that your intermediate integral is wrong (although actually you still got the right answer for the next stage by ignoring the error)

-1/2 cos (2 theta) is :

theta = 0 : -1/2 x 1 = -0.5

theta = pi/2 : -1/2 x -1 = +0.5

so the integral is 0.5 – (-0.5) = 1.0

of course then your next integral starts to make sense as the integral of one with respect to phi is indeed phi. (the integral of zero is zero over all ranges, so clearly otherwise this makes no sense)

(n + 6) / (8 * pi) seems good since it is conservative and is within 5% of your normalization constant for n >= 10, but the error ramps up to 25% at n = 0. It looks like it produces a better error on average (n between 0 and 100) than ( n + 8 ) / (8 * pi), but a worse worst case. I suppose the effect of using this would mean duller than expected specular reflections at very low specular powers.

I’ll update the link to your derivation. Thanks for the help.

The L=N variant is really the only special case of Blinn-Phong that’s reasonably easy to write down in terms of angles analytically. For general L, the plane spanned by L and V (which contains H) doesn’t contain N, so there’s no easy way to write (N.H) in terms of the angles you know; I’d like to see the (n+8)/8pi derivation because I’d very much like to see how that issue is treated there.

Feel free to link to my derivation; however, I clean out my /stuff directory more or less regularly, so this is not a good candidate for a permanent link. I’ve put up a copy of the derivation at (same URL)/articles/phong.pdf where it’s safe.

I wonder about the explanation of (n+8/8pi) even more now. I’d be interested in hearing what Naty says about this when he gets back to you. I expect I’ll meet him at GDC since we’re both Activision employees now, so I’ll try and bug him about this if I don’t hear anything.

I used Maxima to graph the difference between the two functions ((n+8)/8pi)/((n+2)(n+4) / (8pi * (2^(-n/2) + n))), and it seems like it peaks very early (n = 8, error is 7.5%) and then asymptotes 0% error. It takes a long time to get down to anything close to 0% error though, so even at n=100, the difference is still 1.8%.

I wonder if the n+8/8*pi normalization factor is just a relatively cheap-to-compute conservative approximation of the version you derived?

I’ll update my initial post with a link to your derivation if you don’t mind.

The actual integration proceeds in a pretty straightforward fashion – one substitution (which yields a surprisingly neat term), one integration by parts to calculate the valuee (which turns it into a big mess), and then a bunch of algebra to simplify it down again (I’ve kept more intermediate steps of algebraic simplifications than I normally would in a printed document, but after all the whole point of this exercise was to have a complete derivation).

Anyway, the whole story’s got a punchline: the normalization factor I got is (n+2)(n+4) / (8pi * (2^(-n/2) + n)), not (n+8)/8pi. If you plot this factor against n, you’ll notice that it really is close to linear in n, but not quite there.

I was pretty surprised about this result to say the least. I cross-checked the integral with Mathematica, and I also wrote a small test program that computes integrals of functions over the hemisphere using stratified Monte Carlo integration to be sure; I didn’t want to rely on the analytical angle-based formulations too much, since it’s very easy to mess up and accidentally take the wrong angle.

The result, using both analytical computations and the Monte Carlo integration test program, were that with n=16, the overall amount of reflected energy with L=N and the (n+8)/8pi normalization term is about 1.06 times L_i, while the (n+2)(n+4)/… variant derived in the PDF is 1.0 to within integration accuracy. Other values of n yield the same qualitative result: (n+8)/8pi is slightly too large.

After finding this out last weekend, I wrote a mail to Naty Hoffman (who apparently did the derivation for Real-Time Rendering 3rd Ed.), but I didn’t get an answer yet, and I wanted to put this online before I lose my notes with the derivation, so here goes.

That derivation was simpler than I thought it was going to be. I didn’t realise it was just for the original Phong model though. It still bugs me that the n+8/8pi hasn’t been explained though.