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Some questions about Marschner's hair shader

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hi,
I'm reading the paper provided by Marschner in siggraph'03 recently and stuck at equations that prevent me to go ahead. Not sure discussing about these is suitable\permit in this forum since many forums only focus on the shader code itself rather than the theory part. Please let me know if other forums more suitable for this discussion, thanks in advance.

The main part that obscuring me most is the fourth part that talking about scattering from cylinder, the derivations make me confusing including:

1. In equation 2: Why the cos term(with power 2) in denominator corresponding to the projected solid angle of specular cone ? Why i need this term ?
2. In equation 3: The deviation provided by paper seems curious: It says the incident ray deviates rt-ri and each internal reflection deviates pi+2rt, doesn't anyone consider the incident ray deviation should be ri-rt and internal reflection deviation should be pi-2rt ?
3. In equation 4: How could i get this from equation 3? Does anyone can provide derivation about this result? And why only one root for p=0(R) and p=1(TT), and three root for p=2(TRT), any reasonable geometry interpretation about 3 TRT path ?
4. The first equation in appendix B pops so suddenly that interrupt my derivation, where it come from ?

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Some questions about Marschner's hair shader Seila Ancientmoon <seilaancientmoon@gmail.com> - 2012-11-18 22:27 -0800

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