Posts Tagged ‘Fresnel equations’

Marschner Shader Part II

In my last post I mentioned two functions that are needed to represent the hair model as depicted in Marschner’s paper.

S = SR + STT + STRT,

Sp = Mp (q i , q r ) x Np (q d , f d ) for P = R, TT, TRT.

M component

This is actually just a probability density function and the best choice here is to use a Gaussian distribution (or normal distribution).

And the M components are as follows:

  • MR (q h ) = g( Beta R , q h Alpha R).
  • MTT (q h ) = g( Beta TT , q h Alpha TT).
  • MTRT (q h ) = g( Beta TRT , q h Alpha TRT).

N component

The N component is actually a bit tricky to compute. Here are all the main steps:

This is done in order to change the index of refraction to 2D physics, so that the optics of a 3D cylindrical fiber may be reduced to the 2D analysis of the optics of its cross-section.

After looking into Snell’s Law we define the indexes of refraction as:

Remember this picture:

We need to find out who the incident angles are, and we can approximate the solution for this equation as:

Fresnel equation is used in order to simulate the reflection model from within the attenuation

  • Find out the absorption factor

This is actually quite straightforward, just:

  • The attenuation factor

This is obtain combining both the reflection and the absorption factor, hence the “Attenuation by absorption and reflection” model from Marschner’s paper.

where the first derivative is

  • The N component (finally)

and the N are

  • NR (q d , f d ) = NP (0, q d , f d ).
  • NTT (q d , f d ) = NP (1, q d , f d ).
  • NTRT (q d , f d ) = NP (2, q d , f d ).

For the last component Marschner proposes a more complex model in order to avoid singularities, but for my implementation I couldn’t tell any improvement so I stuck with the simpler version of NTRT.

The whole model

As a sum up this is the whole Marschner hair model in just an equation:

Hope I managed to keep everything simple and explicit alike.