Would somebody help me with equation 14.1 ?

[Un Flic]

Hello,

I have hard time wrapping my mind around the derivation of equation 14.1

I understand that that D(Wh) is divided by p(phi) to arrive with pi(theta). I also understand that p(theta) is multiplied by sin(theta) to transform it into spherical coordinates.

However, I don't understand why cos(theta) is raised to power of 3.

I also see in the PDF code that this multiplication by cos(theta) is carried over.

Some explanation of this would be very helpful. A reference for learning more about sampling microfacet models would be even more helpful.

Regards

[Greg Humphreys]

I believe the reason it's raised to the third power is because you end up multiplying by sin(theta) cos(theta) when you convert to spherical coordinates, so that cancels out one of the cos^4 powers in the original Beckmann-Spizzichino model (equation 8.9), and also explains the presence of the sin(theta) term in the numerator.

I agree that this derivation could be made more clear than just simply saying "we'll consider it expressed in spherical coordinates" :)

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Greg Humphreys

ɯoɔ˙lıɐɯƃ@ɹǝdɯnɥ

[Un Flic]

Hello Greg,

Thank you for your input. I think I figured it out, thanks to "Microfacet Models for Refraction through Rough Surfaces" by Bruce Walter, Stephen R. Marschner, Hongsong Li, and Kenneth E. Torrance.

They state explicitly that the sampling is done from D(W_h)cos(theta_h). That makes sense because this cosine factor is needed to make the PDF integrate to 1. In the book this formula is mentioned in 8.4.2.

And then it goes as I described it above - the results from 13.5.3 are used to convert it to spherical coordinates and then formula 13.12 is applied to arrive with PDF for theta.

Well, math is not my strong side all this could be wrong but sounds convincing enough for me to calm me down:)

Regards,