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.TL
Behavior of Materials in RADIANCE
.AU
Greg Ward
.br
Lawrence Berkeley Laboratory
.NH 1
Definitions
.LP
This document describes in gory detail how each material type in RADIANCE
behaves in terms of its parameter values.
The following variables are given:
.LP
.vs 14
.nr VS 14
.TS
center;
l8 l.
$bold R ( P vec , v hat )$	Value of a ray starting at $P vec$ in direction $v hat$ (in watts/sr/m^2)
$P vec sub o$	Eye ray origin
$v hat$	Eye ray direction
$P vec sub s$	Intersection point of ray with surface
$n hat$	Unperturbed surface normal at intersection
$a sub n$	Real argument number $n$
$bold C$	Material color, $"{" a sub 1 , a sub 2 , a sub 3 "}"$
$bold p$	Material pattern value, $"{" 1, 1, 1 "}"$ if none
$d vec$	Material texture vector, $[ 0, 0, 0 ]$ if none
$b vec$	Orientation vector given by the string arguments of anisotropic types
$n sub 1$	Index of refraction on eye ray side
$n sub 2$	Index of refraction on opposite side
$bold A$	Indirect irradiance (in watts/m^2)
$bold A sub t$	Indirect irradiance from opposite side (in watts/m^2)
$m hat$	Mirror direction, which can vary with Monte Carlo sampling
$t sub s$	Specular threshold (set by -st option)
$bold B sub i$	Radiance of light source sample $i$ (in watts/sr/m^2)
$q hat sub i$	Direction to source sample $i$
$omega sub i$	Solid angle of source sample $i$ (in sr)
.TE
.vs 12
.nr VS 12
.LP
Variables with an arrow over them are vectors.
Variables with a circumflex are unit vectors (ie. normalized).
All variables written in bold represent color values.
.NH 2
Derived Variables
.LP
The following values are computed from the variables above:
.LP
.vs 14
.nr VS 14
.TS
center;
l8 l.
$cos sub 1$	Cosine of angle between surface normal and eye ray
$cos sub 2$	Cosine of angle between surface normal and transmitted direction
$n hat sub p$	Perturbed surface normal (after texture application)
$h vec sub i$	Bisecting vector between eye ray and source sample $i$
$F sub TE$	Fresnel coefficient for $TE$-polarized light
$F sub TM$	Fresnel coefficient for $TM$-polarized light
$F$	Fresnel coefficient for unpolarized light
.TE
.vs 12
.nr VS 12
.LP
These values are computed as follows:
.EQ I
cos sub 1	mark ==~ - v hat cdot n hat sub p
.EN
.EQ I
cos sub 2	lineup ==~ sqrt { 1~-~( n sub 1 / n sub 2 )
  sup 2 (1~-~cos sub 1 sup 2 )}
.EN
.EQ I
n hat sub p	lineup ==~ { n hat ~+~ d vec } over { ||~ n hat ~+~ d vec ~|| }
.EN
.EQ I
h vec sub i	lineup ==~ q hat sub i ~-~ v hat
.EN
.EQ I
F sub TE	lineup ==~ left [ { n sub 1 cos sub 1
~-~ n sub 2 cos sub 2 } over { n sub 1 cos sub 1
~+~ n sub 2 cos sub 2 } right ] sup 2
.EN
.EQ I
F sub TM	lineup ==~ left [ {n sub 1 / cos sub 1
~-~ n sub 2 / cos sub 2} over {n sub 1 / cos sub 1
~+~ n sub 2 / cos sub 2} right ] sup 2
.EN
.EQ I
F	lineup ==~ 1 over 2 F sub TE ~+~ 1 over 2 F sub TM
.EN
.NH 2
Vector Math
.LP
Variables that represent vector values are written with an arrow above
(eg. $ v vec $).
Unit vectors (ie. vectors whose lengths are normalized to 1) have a hat
(eg. $ v hat $).
Equations containing vectors are implicitly repeated three times,
once for each component.
Thus, the equation:
.EQ I
v vec ~=~ 2 n hat ~+~ P vec
.EN
is equivalent to the three equations:
.EQ I
v sub x ~=~ 2 n sub x ~+~ P sub x
.EN
.EQ I
v sub y ~=~ 2 n sub y ~+~ P sub y
.EN
.EQ I
v sub z ~=~ 2 n sub z ~+~ P sub z
.EN
There are also cross and dot product operators defined for vectors, as
well as the vector norm:
.RS
.LP
.UL "Vector Dot Product"
.EQ I
a vec cdot b vec ~==~ a sub x b sub x ~+~ a sub y b sub y ~+~ a sub z b sub z
.EN
.LP
.UL "Vector Cross Product"
.EQ I
a vec ~ times ~ b vec ~==~ left | matrix {
  ccol { i hat above a sub x above b sub x }
  ccol { j hat above a sub y above b sub y }
  ccol { k hat above a sub z above b sub z }
} right |
.EN
or, written out:
.EQ I
a vec ~ times ~ b vec ~=~ [ a sub y b sub z ~-~ a sub z b sub y ,~
a sub z b sub x ~-~ a sub x b sub z ,~ a sub x b sub y ~-~ a sub y b sub x ]
.EN
.LP
.UL "Vector Norm"
.EQ I
|| v vec || ~==~ sqrt {{v sub x} sup 2 ~+~ {v sub y} sup 2 ~+~ {v sub z} sup 2}
.EN
.RE
.LP
Values are collected into a vector using square brackets:
.EQ I
v vec ~=~ [ v sub x , v sub y , v sub z ]
.EN
.NH 2
Color Math
.LP
Variables that represent color values are written in boldface type.
Color values may have any number of spectral samples.
Currently, RADIANCE uses only three such values, referred to generically
as red, green and blue.
Whenever a color variable appears in an equation, that equation is
implicitly repeated once for each spectral sample.
Thus, the equation:
.EQ I
bold C ~=~ bold A^bold B ~+~ d bold F
.EN
is shorthand for the set of equations:
.EQ I
C sub 1 ~=~ A sub 1 B sub 1 ~+~ d F sub 1
.EN
.EQ I
C sub 2 ~=~ A sub 2 B sub 2 ~+~ d F sub 2
.EN
.EQ I
C sub 3 ~=~ A sub 3 B sub 3 ~+~ d F sub 3
.EN
	...
.LP
And so on for however many spectral samples are used.
Note that color math is not the same as vector math.
For example, there is no such thing as
a dot product for colors.
.LP
Curly braces are used to collect values into a single color, like so:
.EQ I
bold C ~=~ "{" r, g, b "}"
.EN
.sp 2
.NH
Light Sources
.LP
Light sources are extremely simple in their behavior when viewed directly.
Their color in a particular direction is given by the equation:
.EQ L
bold R ~=~ bold p^bold C
.EN
.LP
The special light source material types, glow, spotlight, and illum,
differ only in their affect on the direct calculation, ie. which rays
are traced to determine shadows.
These differences are explained in the RADIANCE reference manual, and
will not be repeated here.
.sp 2
.NH
Specular Types
.LP
Specular material types do not involve special light source testing and
are thus are simpler to describe than surfaces with a diffuse component.
The output radiance is usually a function of one or two other ray 
evaluations.
.NH 2
Mirror
.LP
The value at a mirror surface is a function of the ray value in
the mirror direction:
.EQ L
bold R ~=~ bold p^bold C^bold R ( P vec sub s ,~ m hat )
.EN
.NH 2
Dieletric
.LP
The value of a dieletric material is computed from Fresnel's equations:
.EQ L
bold R ~=~ bold p^bold C sub t ( 1 - F )^bold R ( P vec sub s ,~ t hat ) ~+~
  bold C sub t^F^bold R ( P vec sub s ,~ m hat )
.EN
where:
.EQ I
bold C sub t ~=~ bold C sup {|| P vec sub s ~-~ P vec sub o ||}
.EN
.EQ I
t hat ~=~ n sub 1 over n sub 2 v hat ~+~ left ( n sub 1 over n sub 2 cos sub 1
  ~-~ cos sub 2 right ) n hat sub p
.EN
.LP
The Hartmann constant is used only to calculate the index of refraction
for a dielectric, and does not otherwise influence the above equations.
In particular, transmitted directions are not sampled based on dispersion.
Dispersion is only modeled in a very crude way when a light source is
casting a beam towards the eye point.
We will not endeavor to explain the algorithm here as it is rather
nasty.
.LP
For the material type "interface", the color which is used for $bold C$
as well as the indices of refraction $n sub 1$ and $n sub 2$ is determined
by which direction the ray is headed.
.NH 2
Glass
.LP
Glass uses an infinite series solution to the interreflection inside a pane
of thin glass.
.EQ L
bold R ~=~ bold p^bold C sub t^bold R ( P vec sub s ,~ t hat )
  left [ 1 over 2 {(1 ~-~ F sub TE )} sup 2 over {1 ~-~ F sub TE sup 2
     bold C sub t sup 2 } ~+~ 1 over 2 {(1 ~-~ F sub TM )} sup 2
     over {1 ~-~ F sub TM sup 2 bold C sub t sup 2 } right ] ~+~
  bold R ( P vec sub s ,~ m hat ) left [ 1 over 2 {F sub TE (1 ~+~
     (1 ~-~ 2 F sub TE ) bold C sub t sup 2 )} over {1 ~-~ F sub TE sup 2
     bold C sub t sup 2 } ~+~ 1 over 2 {F sub TM (1 ~+~
     (1 ~-~ 2 F sub TM ) bold C sub t sup 2 )} over {1 ~-~ F sub TM sup 2
     bold C sub t sup 2 } right ]
.EN
where:
.EQ I
bold C sub t ~=~ bold C sup {(1/ cos sub 2 )}
.EN
.EQ I
t hat ~=~ v hat~+~2(1^-^n sub 2 ) d vec
.EN
.sp 2
.NH
Basic Reflection Model
.LP
The basic reflection model used in RADIANCE takes into account both specular
and diffuse interactions with both sides of a surface.
Most RADIANCE material types are special cases of this more general formula:
.EQ L (1)
bold R ~=~ mark size +3 sum from sources bold B sub i omega sub i
  left { Max(0,~ q hat sub i cdot n hat sub p )
    left ( bold rho sub d over pi ~+~ bold rho sub si right ) ~+~
    Max(0,~ - q hat sub i cdot n hat sub p )
    left ( bold tau sub d over pi ~+~ bold tau sub si right ) right }
.EN
.EQ L
lineup ~~+~~ bold rho sub s bold R ( P vec sub s ,~ m hat ) ~~+~~
  bold tau sub s bold R ( P vec sub s ,~ t hat )
.EN
.EQ L
lineup ~~+~~ bold rho sub a over pi bold A ~~+~~
  bold tau sub a over pi bold A sub t
.EN
Note that only one of the transmitted or reflected components in the first
term of the above equation can be non-zero, depending on whether the given
light source is in front of or behind the surface.
The values of the various $ bold rho $ and $ bold tau $ variables will be
defined differently for each material type, and are given in the following
sections for plastic, metal and trans.
.NH 2
Plastic
.LP
A plastic surface has uncolored highlights and no transmitted component.
If the surface roughness ($a sub 5$) is zero or the specularity
($bold r sub s$) is greater than the threshold ($t sub s$)
then a ray is traced in or near the mirror direction.
An approximation to the Fresnel reflection coefficient
($bold r sub s ~approx~ 1 - F$) is used to modify the specularity to account
for the increase in specular reflection near grazing angles.
.LP
The reflection formula for plastic is obtained by adding the following
definitions to the basic formula given in equation (1):
.EQ I
bold rho sub d ~=~ bold p^bold C (1 ~-~ a sub 4 )
.EN
.EQ I
bold rho sub si ~=~ left {~ lpile {{bold r sub s
  {f sub s ( q hat sub i )} over
  {( q hat sub i cdot n hat sub p ) cos sub 1}} above
  0 } ~~~ lpile { {if~a sub 5 >0} above otherwise }
.EN
.EQ I
bold rho sub s ~=~ left {~ lpile {{bold r sub s} above 0 } ~~~
  lpile {{if~a sub 5^=^0~or~bold r sub s^>^t sub s} above otherwise }
.EN
.EQ I
bold rho sub a ~=~ left {~ lpile {{ bold p^bold C^(1~-~bold r sub s )} above
  {bold p^bold C}} ~~~ lpile
  {{if~a sub 5^=^0~or~bold r sub s^>^t sub s} above otherwise }
.EN
.EQ I
bold tau sub a ,~ bold tau sub d ,~ bold tau sub si ,~ bold tau sub s ~=~ 0
.EN
.EQ I
bold r sub s ~=~ a sub 4
.EN
.EQ I
f sub s ( q hat sub i ) ~=~ e sup{[ ( h vec sub i cdot n hat sub p ) sup 2
  ~-~ || h vec || sup 2 ]/
  ( h vec sub i cdot n hat sub p ) sup 2 / alpha sub i}
  over {4 pi alpha sub i}
.EN
.EQ I
alpha sub i ~=~ a sub 5 sup 2 ~+~ omega sub i over {4 pi}
.EN
.LP
There is one additional caveat to the above formulas.
If the roughness is greater than zero and the reflected ray,
$bold R ( P vec sub s ,~ r hat )$,
intersects a light source, then it is not used in the calculation.
Using such a ray would constitute double-counting, since the direct
component has already been included in the source sample summation.
.NH 2
Metal
.LP
Metal is identical to plastic, except for the definition of $ bold r sub s $,
which now includes the color of the material:
.EQ I
bold r sub s ~=~
  "{" a sub 1 a sub 4 ,~
  a sub 2 a sub 4 ,~
  a sub 3 a sub 4 "}"
.EN
.NH 2
Trans
.LP
The trans type adds transmitted and colored specular and diffuse components
to the colored diffuse and uncolored specular components of the plastic type.
Again, the roughness value and specular threshold determine
whether or not specular rays will be followed for this material.
.EQ I
bold rho sub d ~=~ bold p^bold C^(1 ~-~ a sub 4 ) ( 1 ~-~ a sub 6 )
.EN
.EQ I
bold rho sub si ~=~ left {~ lpile {{bold r sub s
  {f sub s ( q hat sub i )} over
  {( q hat sub i cdot n hat sub p ) cos sub 1}} above
  0 } ~~~ lpile { {if~a sub 5 >0} above otherwise }
.EN
.EQ I
bold rho sub s ~=~ left {~ lpile {{bold r sub s} above 0 } ~~~
  lpile {{if~a sub 5^=^0~or~bold r sub s^>^t sub s} above otherwise }
.EN
.EQ I
bold rho sub a ~=~ left {~ lpile {{ bold p^bold C^
  (1~-~bold r sub s ) (1~-~a sub 6 )} above
  {bold p^bold C^(1~-~a sub 6 )}} ~~~ lpile
  {{if~a sub 5^=^0~or~bold r sub s^>^t sub s} above otherwise }
.EN
.EQ I
bold tau sub d ~=~ a sub 6 (1~-~bold r sub s ) (1~-~a sub 7 )^bold p^bold C
.EN
.EQ I
bold tau sub si ~=~ left {~ lpile {{a sub 6 a sub 7 (1~-~bold r sub s )^
  bold p^bold C {g sub s ( q hat sub i )} over
  {(- q hat sub i cdot n hat sub p ) cos sub 1}} above
  0 } ~~~ lpile { {if~a sub 5^>^0} above otherwise }
.EN
.EQ I
bold tau sub s ~=~ left {~ lpile {{a sub 6 a sub 7 (1~-~bold r sub s )^
  bold p^bold C} above 0} ~~~ lpile
  {{if~a sub 5^=^0~or~a sub 6 a sub 7 (1~-~bold r sub s )^>^t sub s}
  above otherwise}
.EN
.EQ I
bold tau sub a ~=~ left {~ lpile {{a sub 6 (1~-~bold r sub s ) (1~-~a sub 7 )
  ^bold p^bold C} above {a sub 6 (1~-~bold r sub s )^bold p^bold C}} ~~~
  lpile {{if~a sub 5^=^0~or~a sub 6 a sub 7 (1~-~bold r sub s )^>^t sub s}
  above otherwise}
.EN
.EQ I
bold r sub s ~=~ a sub 4
.EN
.EQ I
f sub s ( q hat sub i ) ~=~ e sup{[ ( h vec sub i cdot n hat sub p ) sup 2
  ~-~ || h vec || sup 2 ]/
  ( h vec sub i cdot n hat sub p ) sup 2 / alpha sub i}
  over {4 pi alpha sub i}
.EN
.EQ I
alpha sub i ~=~ a sub 5 sup 2 ~+~ omega sub i over {4 pi}
.EN
.EQ I
g sub s ( q hat sub i ) ~=~ e sup{( 2 q hat sub i cdot t hat~-~2)/ beta sub i}
  over {pi beta sub i}
.EN
.EQ I
t hat ~=~ {v hat~-~d vec}over{|| v hat~-~d vec ||}
.EN
.EQ I
beta sub i ~=~ a sub 5 sup 2 ~+~ omega sub i over pi
.EN
.LP
The same caveat applies to specular rays generated for trans type as did
for plastic and metal.
Namely, if the roughness is greater than zero and the reflected ray,
$bold R ( P vec sub s ,~ r hat )$,
or the transmitted ray,
$bold R ( P vec sub s ,~ t hat )$,
intersects a light source, then it is not used in the calculation.
.NH 2
Anisotropic Types
.LP
The anisotropic reflectance types (plastic2, metal2, trans2) use the
same formulas as their counterparts with the exception of the
exponent terms, $f sub s ( q hat sub i )$ and
$g sub s ( q hat sub i )$.
These terms now use an additional vector, $b vec$, to
orient an elliptical highlight.
(Note also that the argument numbers for the type trans2 have been
changed so that $a sub 6$ is $a sub 7$ and $a sub 7$ is $a sub 8$.)\0
.EQ I
f sub s ( q hat sub i ) ~=~
  1 over {4 pi sqrt {alpha sub ix alpha sub iy} }
  exp left [ -^{{( h vec sub i cdot x hat )} sup 2 over{alpha sub ix}
  ~+~ {( h vec sub i cdot y hat )} sup 2 over{alpha sub iy}} over
  {( h vec sub i cdot n hat sub p ) sup 2}right ]
.EN
.EQ I
x hat ~=~ y hat~times~n hat sub p
.EN
.EQ I
y hat ~=~ {n hat sub p~times~b vec}over{|| n hat sub p~times~b vec ||}
.EN
.EQ I
alpha sub ix ~=~ a sub 5 sup 2 ~+~ omega sub i over {4 pi}
.EN
.EQ I
alpha sub iy ~=~ a sub 6 sup 2 ~+~ omega sub i over {4 pi}
.EN
.EQ I
g sub s ( q hat sub i ) ~=~
  1 over {pi sqrt {beta sub ix beta sub iy} }
  exp left [ {{( c vec sub i cdot x hat )} sup 2 over{beta sub ix}
  ~+~ {( c vec sub i cdot y hat )} sup 2 over{beta sub iy}} over
  {{( n hat sub p cdot c vec sub i )}sup 2 over
  {|| c vec sub i ||}sup 2 ~-~ 1}
  right ]
.EN
.EQ I
c vec sub i ~=~ q hat sub i~-~t hat
.EN
.EQ I
t hat ~=~ {v hat~-~d vec}over{|| v hat~-~d vec ||}
.EN
.EQ I
beta sub ix ~=~ a sub 5 sup 2 ~+~ omega sub i over pi
.EN
.EQ I
beta sub iy ~=~ a sub 6 sup 2 ~+~ omega sub i over pi
.EN
.NH 2
BRDF Types
.LP
The more general brdf types (plasfunc, plasdata, metfunc, metdata,
BRTDfunc) use the same basic formula given in equation (1),
but allow the user to specify $bold rho sub si$ and $bold tau sub si$ as
either functions or data, instead of using the default
Gaussian formulas.
Note that only the exponent terms, $f sub s ( q hat sub i )$ and
$g sub s ( q hat sub i )$ with the radicals in their denominators
are replaced, and not the coefficients preceding them.
It is very important that the user give properly normalized functions (ie.
functions that integrate to 1 over the hemisphere) to maintain correct
energy balance.