Here are some notes for the GGX paper "Microfacet Models for Refraction through Rough Surfaces". In this article, I will give derivations for some important equation in this paper.
Derivation for equation (8)
This equation tell us how to construct a macrosurface BRDF given microsurface 's D, G, F.
In this equation,
is
incident vector.
is
outgoing vector.
is
macrosurface normal.
is
microsurface normal.
is
the macrosurface BRDF.
is
the microsurface BRDF.
is
microfacet distribution function.
is
shadowing-masking function.
is
solid angle in the hemisphere 
.
How can we get this equation? Please see the figure below.
In this figure, the surface is illuminated
by a light source and an observer is looking at the surface. The observer has a
microscope so that he will see the microfacets s1, s2,
... . These microfacets have different colors because of their orientations are
different. When the observer look at the surface at the same location, but
without the microscope, he will no longer see the microfacets but a uniform color.
This time, he knows that it is colors from microfacets that mix together and
form the uniform color. Let's denote the colors from microfacets as 
and
the uniform color as 
.
Then we have:
In this equation,
is
the projected area of si . According to the definition of microfacet
distribution function, we have:
In this equation,
is
the area of macrosurface.
is i-th microfacet's normal.
is
a small solid angle aligned with .
is
microfacet distribution function.
Combine these equations, we have:
We can
eliminate and using the equation
(3) in the paper:
Convert sum to integral, then we get:
Now we
can see the term 
.
Let's investigate the term
further:
In this equation,
is
a small solid angle of incident light over the hemisphere .
Finally, we have:
According to the rendering equation
We can
regard the inner integral 
as
the equivalent BRDF for macrosurface, so that we get:
Confirm equation (9)
According to the definition of radiance:
In this equation,
is
the radiance in outgoing direction
is
luminous flux
is area of
macrofacet
So the outgoing irradiance is:
Put equation (9) in, we have:
According to equation (10),
So According to equation (9), the overall
outgoing irradiance equals the incoming irradiance scaled by a factor 
,
which is less than 1.
Derivation for equation (20)
According to equation (8),
Put equation (15) in it, we have:
When 
,
,
then according to equation (10), we have:
Derivation for equation (42)
Let us take a careful look at the
definition of 
.
Suppose there is one point 
on
surface whose normal is ,
we construct a plane
perpendicular
to the normal and choose two perpendicular axes 
and
.
For a small patch 
on the plane,
we denote the direction pointing from to
it ,
and the small solid angle it occupies 
.
According to equation (4), which is
We can consider 
as the probability density 
of
finding a microfacet whose normal is
inside ,
so that we have
That's exactly what is.
Derivation for equation (45)
Break the ray into many short segments,
each with projected length 
.
According to the paper, the probability that the ray is first blocked in
segment 
is
, so the probability that ray is always unblocked is:
Then we have:
From calculus we know:
So
So that
Derivation for equation (46)
Let's consider the situation that a ray
intersects with an short and straight surface segment .
In order to do that, the surface height should below the ray at 
and
above the ray at 
. For a given slope 
,
there exist a set of surface segments that fulfill this condition, which are in
the shaded areas in the figures below.
It's easy to note that the possible surface
height at varies from 
to
.
So given a surface with slope ,
the probability that it intersects with a ray is
In this equation,
is
the probability density that surface height reaches at
point .
Also we know that the probability that a
surface segment has slope is
Combine them, then we get the probability
for finding a surface segment with slope as
well as intersecting with the ray:
Consider all possible surface slope ,
the probability that they intersect with the ray is:
Among all surface segments with slope ,
the probability that a surface segment below the ray is:
Consider all possible surface slope ,
the probability that they below the ray is:
Assume 
is
independent from 
,
we have:
So the probability that a ray first
intersects with surface in 
is:
Let 
,
so
Derivation for equation (49)
From equation (48), we have:
Note that
So
Then
Put it in equation (45):
Derivation for equation (50)
Explanation of Beckmann Distribution Model
The distribution function of Beckmann Model is:
Let’s see the meaning of Beckmann
Distribution Model. I use the way when I derive equation (42). That is, make a
plane perpendicular to the surface normal, and make sure the distance between
the plane and surface point O is 1. For a microfacet normal ,
it must be pointing at a point P on the plane. So we can transform the
probability density of to
another density function on the plane. I will show that it is a Gaussian
function.
According to equation (4), which is
We can consider as the probability density of finding
a microfacet whose normal is
inside .
Also, we know:
In this equation, 
is
the distance between projected point P and surface point O.
So the probability of finding a microfacet
whose normal is
inside is:
Look! It’s a gaussian distribution function
with variation 
.
That’s what Beckmann Distribution really is.
G1 for Beckmann Distribution
According to equation (51),
where
For Beckmann Distribution,
Also we know 
,
so
From normal distribution we know:
So
The first integration is:
The second one is:
Put 
and
in
it, we have:
Finally,
Explanation of Phong Distribution Model
The distribution function of Phong model is:
The same as Beckmann model, we try to
project 
to
a plane above the surface point P, that is
We can confirm that the overall probability is
