ray/doc/materials.1

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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.
Revision:	1.3
Committed:	Tue Sep 21 18:44:09 2004 UTC (21 years, 1 month ago) by greg
Branch:	MAIN
CVS Tags:	rad5R4, rad5R2, rad5R3, rad5R0, rad5R1, rad4R2, rad3R7P2, rad3R7P1, rad6R0, rad4R1, rad4R0, rad3R6, rad3R6P1, rad3R8, rad3R9, rad4R2P1, rad4R2P2, HEAD
Changes since 1.2:	+4 -4 lines
Log Message:	Fixed RCS Id and grouping of terms from last change for BRDF
#	User	Rev	Content
1	greg	1.3	.\" RCSid "$Id$"
2	greg	1.1	.\"Print with tbl and eqn or neqn with -ms macro package
3			.vs 12
4			.nr VS 12
5			.nr PD .5v \" paragraph distance (inter-paragraph spacing)
6			.EQ L
7			delim $$
8			.EN
9			.DA
10			.TL
11			Behavior of Materials in RADIANCE
12			.AU
13			Greg Ward
14			.br
15			Lawrence Berkeley Laboratory
16			.NH 1
17			Definitions
18			.LP
19			This document describes in gory detail how each material type in RADIANCE
20			behaves in terms of its parameter values.
21			The following variables are given:
22			.LP
23			.vs 14
24			.nr VS 14
25			.TS
26			center;
27			l8 l.
28			$bold R ( P vec , v hat )$ Value of a ray starting at $P vec$ in direction $v hat$ (in watts/sr/m^2)
29			$P vec sub o$ Eye ray origin
30			$v hat$ Eye ray direction
31			$P vec sub s$ Intersection point of ray with surface
32			$n hat$ Unperturbed surface normal at intersection
33			$a sub n$ Real argument number $n$
34			$bold C$ Material color, $"{" a sub 1 , a sub 2 , a sub 3 "}"$
35			$bold p$ Material pattern value, $"{" 1, 1, 1 "}"$ if none
36			$d vec$ Material texture vector, $[ 0, 0, 0 ]$ if none
37			$b vec$ Orientation vector given by the string arguments of anisotropic types
38			$n sub 1$ Index of refraction on eye ray side
39			$n sub 2$ Index of refraction on opposite side
40			$bold A$ Indirect irradiance (in watts/m^2)
41			$bold A sub t$ Indirect irradiance from opposite side (in watts/m^2)
42			$m hat$ Mirror direction, which can vary with Monte Carlo sampling
43			$t sub s$ Specular threshold (set by -st option)
44			$bold B sub i$ Radiance of light source sample $i$ (in watts/sr/m^2)
45			$q hat sub i$ Direction to source sample $i$
46			$omega sub i$ Solid angle of source sample $i$ (in sr)
47			.TE
48			.vs 12
49			.nr VS 12
50			.LP
51			Variables with an arrow over them are vectors.
52			Variables with a circumflex are unit vectors (ie. normalized).
53			All variables written in bold represent color values.
54			.NH 2
55			Derived Variables
56			.LP
57			The following values are computed from the variables above:
58			.LP
59			.vs 14
60			.nr VS 14
61			.TS
62			center;
63			l8 l.
64			$cos sub 1$ Cosine of angle between surface normal and eye ray
65			$cos sub 2$ Cosine of angle between surface normal and transmitted direction
66			$n hat sub p$ Perturbed surface normal (after texture application)
67			$h vec sub i$ Bisecting vector between eye ray and source sample $i$
68			$F sub TE$ Fresnel coefficient for $TE$-polarized light
69			$F sub TM$ Fresnel coefficient for $TM$-polarized light
70			$F$ Fresnel coefficient for unpolarized light
71			.TE
72			.vs 12
73			.nr VS 12
74			.LP
75			These values are computed as follows:
76			.EQ I
77			cos sub 1 mark ==~ - v hat cdot n hat sub p
78			.EN
79			.EQ I
80			cos sub 2 lineup ==~ sqrt { 1~-~( n sub 1 / n sub 2 )
81			sup 2 (1~-~cos sub 1 sup 2 )}
82			.EN
83			.EQ I
84			n hat sub p lineup ==~ { n hat ~+~ d vec } over { \|\|~ n hat ~+~ d vec ~\|\| }
85			.EN
86			.EQ I
87			h vec sub i lineup ==~ q hat sub i ~-~ v hat
88			.EN
89			.EQ I
90			F sub TE lineup ==~ left [ { n sub 1 cos sub 1
91			~-~ n sub 2 cos sub 2 } over { n sub 1 cos sub 1
92			~+~ n sub 2 cos sub 2 } right ] sup 2
93			.EN
94			.EQ I
95			F sub TM lineup ==~ left [ {n sub 1 / cos sub 1
96			~-~ n sub 2 / cos sub 2} over {n sub 1 / cos sub 1
97			~+~ n sub 2 / cos sub 2} right ] sup 2
98			.EN
99			.EQ I
100			F lineup ==~ 1 over 2 F sub TE ~+~ 1 over 2 F sub TM
101			.EN
102			.NH 2
103			Vector Math
104			.LP
105			Variables that represent vector values are written with an arrow above
106			(eg. $ v vec $).
107			Unit vectors (ie. vectors whose lengths are normalized to 1) have a hat
108			(eg. $ v hat $).
109			Equations containing vectors are implicitly repeated three times,
110			once for each component.
111			Thus, the equation:
112			.EQ I
113			v vec ~=~ 2 n hat ~+~ P vec
114			.EN
115			is equivalent to the three equations:
116			.EQ I
117			v sub x ~=~ 2 n sub x ~+~ P sub x
118			.EN
119			.EQ I
120			v sub y ~=~ 2 n sub y ~+~ P sub y
121			.EN
122			.EQ I
123			v sub z ~=~ 2 n sub z ~+~ P sub z
124			.EN
125			There are also cross and dot product operators defined for vectors, as
126			well as the vector norm:
127			.RS
128			.LP
129			.UL "Vector Dot Product"
130			.EQ I
131			a vec cdot b vec ~==~ a sub x b sub x ~+~ a sub y b sub y ~+~ a sub z b sub z
132			.EN
133			.LP
134			.UL "Vector Cross Product"
135			.EQ I
136			a vec ~ times ~ b vec ~==~ left \| matrix {
137			ccol { i hat above a sub x above b sub x }
138			ccol { j hat above a sub y above b sub y }
139			ccol { k hat above a sub z above b sub z }
140			} right \|
141			.EN
142			or, written out:
143			.EQ I
144			a vec ~ times ~ b vec ~=~ [ a sub y b sub z ~-~ a sub z b sub y ,~
145			a sub z b sub x ~-~ a sub x b sub z ,~ a sub x b sub y ~-~ a sub y b sub x ]
146			.EN
147			.LP
148			.UL "Vector Norm"
149			.EQ I
150			\|\| v vec \|\| ~==~ sqrt {{v sub x} sup 2 ~+~ {v sub y} sup 2 ~+~ {v sub z} sup 2}
151			.EN
152			.RE
153			.LP
154			Values are collected into a vector using square brackets:
155			.EQ I
156			v vec ~=~ [ v sub x , v sub y , v sub z ]
157			.EN
158			.NH 2
159			Color Math
160			.LP
161			Variables that represent color values are written in boldface type.
162			Color values may have any number of spectral samples.
163			Currently, RADIANCE uses only three such values, referred to generically
164			as red, green and blue.
165			Whenever a color variable appears in an equation, that equation is
166			implicitly repeated once for each spectral sample.
167			Thus, the equation:
168			.EQ I
169			bold C ~=~ bold A^bold B ~+~ d bold F
170			.EN
171			is shorthand for the set of equations:
172			.EQ I
173			C sub 1 ~=~ A sub 1 B sub 1 ~+~ d F sub 1
174			.EN
175			.EQ I
176			C sub 2 ~=~ A sub 2 B sub 2 ~+~ d F sub 2
177			.EN
178			.EQ I
179			C sub 3 ~=~ A sub 3 B sub 3 ~+~ d F sub 3
180			.EN
181			...
182			.LP
183			And so on for however many spectral samples are used.
184			Note that color math is not the same as vector math.
185			For example, there is no such thing as
186			a dot product for colors.
187			.LP
188			Curly braces are used to collect values into a single color, like so:
189			.EQ I
190			bold C ~=~ "{" r, g, b "}"
191			.EN
192			.sp 2
193			.NH
194			Light Sources
195			.LP
196			Light sources are extremely simple in their behavior when viewed directly.
197			Their color in a particular direction is given by the equation:
198			.EQ L
199			bold R ~=~ bold p^bold C
200			.EN
201			.LP
202			The special light source material types, glow, spotlight, and illum,
203			differ only in their affect on the direct calculation, ie. which rays
204			are traced to determine shadows.
205			These differences are explained in the RADIANCE reference manual, and
206			will not be repeated here.
207			.sp 2
208			.NH
209			Specular Types
210			.LP
211			Specular material types do not involve special light source testing and
212			are thus are simpler to describe than surfaces with a diffuse component.
213			The output radiance is usually a function of one or two other ray
214			evaluations.
215			.NH 2
216			Mirror
217			.LP
218			The value at a mirror surface is a function of the ray value in
219			the mirror direction:
220			.EQ L
221			bold R ~=~ bold p^bold C^bold R ( P vec sub s ,~ m hat )
222			.EN
223			.NH 2
224			Dieletric
225			.LP
226			The value of a dieletric material is computed from Fresnel's equations:
227			.EQ L
228			bold R ~=~ bold p^bold C sub t ( 1 - F )^bold R ( P vec sub s ,~ t hat ) ~+~
229			bold C sub t^F^bold R ( P vec sub s ,~ m hat )
230			.EN
231			where:
232			.EQ I
233			bold C sub t ~=~ bold C sup {\|\| P vec sub s ~-~ P vec sub o \|\|}
234			.EN
235			.EQ I
236			t hat ~=~ n sub 1 over n sub 2 v hat ~+~ left ( n sub 1 over n sub 2 cos sub 1
237			~-~ cos sub 2 right ) n hat sub p
238			.EN
239			.LP
240			The Hartmann constant is used only to calculate the index of refraction
241			for a dielectric, and does not otherwise influence the above equations.
242			In particular, transmitted directions are not sampled based on dispersion.
243			Dispersion is only modeled in a very crude way when a light source is
244			casting a beam towards the eye point.
245			We will not endeavor to explain the algorithm here as it is rather
246			nasty.
247			.LP
248			For the material type "interface", the color which is used for $bold C$
249			as well as the indices of refraction $n sub 1$ and $n sub 2$ is determined
250			by which direction the ray is headed.
251			.NH 2
252			Glass
253			.LP
254			Glass uses an infinite series solution to the interreflection inside a pane
255			of thin glass.
256			.EQ L
257			bold R ~=~ bold p^bold C sub t^bold R ( P vec sub s ,~ t hat )
258			left [ 1 over 2 {(1 ~-~ F sub TE )} sup 2 over {1 ~-~ F sub TE sup 2
259			bold C sub t sup 2 } ~+~ 1 over 2 {(1 ~-~ F sub TM )} sup 2
260			over {1 ~-~ F sub TM sup 2 bold C sub t sup 2 } right ] ~+~
261			bold R ( P vec sub s ,~ m hat ) left [ 1 over 2 {F sub TE (1 ~+~
262			(1 ~-~ 2 F sub TE ) bold C sub t sup 2 )} over {1 ~-~ F sub TE sup 2
263			bold C sub t sup 2 } ~+~ 1 over 2 {F sub TM (1 ~+~
264			(1 ~-~ 2 F sub TM ) bold C sub t sup 2 )} over {1 ~-~ F sub TM sup 2
265			bold C sub t sup 2 } right ]
266			.EN
267			where:
268			.EQ I
269			bold C sub t ~=~ bold C sup {(1/ cos sub 2 )}
270			.EN
271			.EQ I
272			t hat ~=~ v hat~+~2(1^-^n sub 2 ) d vec
273			.EN
274			.sp 2
275			.NH
276			Basic Reflection Model
277			.LP
278			The basic reflection model used in RADIANCE takes into account both specular
279			and diffuse interactions with both sides of a surface.
280			Most RADIANCE material types are special cases of this more general formula:
281			.EQ L (1)
282			bold R ~=~ mark size +3 sum from sources bold B sub i omega sub i
283			left { Max(0,~ q hat sub i cdot n hat sub p )
284			left ( bold rho sub d over pi ~+~ bold rho sub si right ) ~+~
285			Max(0,~ - q hat sub i cdot n hat sub p )
286			left ( bold tau sub d over pi ~+~ bold tau sub si right ) right }
287			.EN
288			.EQ L
289			lineup ~~+~~ bold rho sub s bold R ( P vec sub s ,~ m hat ) ~~+~~
290			bold tau sub s bold R ( P vec sub s ,~ t hat )
291			.EN
292			.EQ L
293			lineup ~~+~~ bold rho sub a over pi bold A ~~+~~
294			bold tau sub a over pi bold A sub t
295			.EN
296			Note that only one of the transmitted or reflected components in the first
297			term of the above equation can be non-zero, depending on whether the given
298			light source is in front of or behind the surface.
299			The values of the various $ bold rho $ and $ bold tau $ variables will be
300			defined differently for each material type, and are given in the following
301			sections for plastic, metal and trans.
302			.NH 2
303			Plastic
304			.LP
305			A plastic surface has uncolored highlights and no transmitted component.
306			If the surface roughness ($a sub 5$) is zero or the specularity
307			($bold r sub s$) is greater than the threshold ($t sub s$)
308			then a ray is traced in or near the mirror direction.
309			An approximation to the Fresnel reflection coefficient
310			($bold r sub s ~approx~ 1 - F$) is used to modify the specularity to account
311			for the increase in specular reflection near grazing angles.
312			.LP
313			The reflection formula for plastic is obtained by adding the following
314			definitions to the basic formula given in equation (1):
315			.EQ I
316			bold rho sub d ~=~ bold p^bold C (1 ~-~ a sub 4 )
317			.EN
318			.EQ I
319			bold rho sub si ~=~ left {~ lpile {{bold r sub s
320			{f sub s ( q hat sub i )} over
321	greg	1.3	{( q hat sub i cdot n hat sub p ) cos sub 1}} above
322	greg	1.1	0 } ~~~ lpile { {if~a sub 5 >0} above otherwise }
323			.EN
324			.EQ I
325			bold rho sub s ~=~ left {~ lpile {{bold r sub s} above 0 } ~~~
326			lpile {{if~a sub 5^=^0~or~bold r sub s^>^t sub s} above otherwise }
327			.EN
328			.EQ I
329			bold rho sub a ~=~ left {~ lpile {{ bold p^bold C^(1~-~bold r sub s )} above
330			{bold p^bold C}} ~~~ lpile
331			{{if~a sub 5^=^0~or~bold r sub s^>^t sub s} above otherwise }
332			.EN
333			.EQ I
334			bold tau sub a ,~ bold tau sub d ,~ bold tau sub si ,~ bold tau sub s ~=~ 0
335			.EN
336			.EQ I
337			bold r sub s ~=~ a sub 4
338			.EN
339			.EQ I
340			f sub s ( q hat sub i ) ~=~ e sup{[ ( h vec sub i cdot n hat sub p ) sup 2
341			~-~ \|\| h vec \|\| sup 2 ]/
342			( h vec sub i cdot n hat sub p ) sup 2 / alpha sub i}
343			over {4 pi alpha sub i}
344			.EN
345			.EQ I
346			alpha sub i ~=~ a sub 5 sup 2 ~+~ omega sub i over {4 pi}
347			.EN
348			.LP
349			There is one additional caveat to the above formulas.
350			If the roughness is greater than zero and the reflected ray,
351			$bold R ( P vec sub s ,~ r hat )$,
352			intersects a light source, then it is not used in the calculation.
353			Using such a ray would constitute double-counting, since the direct
354			component has already been included in the source sample summation.
355			.NH 2
356			Metal
357			.LP
358			Metal is identical to plastic, except for the definition of $ bold r sub s $,
359			which now includes the color of the material:
360			.EQ I
361			bold r sub s ~=~
362			"{" a sub 1 a sub 4 ,~
363			a sub 2 a sub 4 ,~
364			a sub 3 a sub 4 "}"
365			.EN
366			.NH 2
367			Trans
368			.LP
369			The trans type adds transmitted and colored specular and diffuse components
370			to the colored diffuse and uncolored specular components of the plastic type.
371			Again, the roughness value and specular threshold determine
372			whether or not specular rays will be followed for this material.
373			.EQ I
374			bold rho sub d ~=~ bold p^bold C^(1 ~-~ a sub 4 ) ( 1 ~-~ a sub 6 )
375			.EN
376			.EQ I
377			bold rho sub si ~=~ left {~ lpile {{bold r sub s
378			{f sub s ( q hat sub i )} over
379	greg	1.3	{( q hat sub i cdot n hat sub p ) cos sub 1}} above
380	greg	1.1	0 } ~~~ lpile { {if~a sub 5 >0} above otherwise }
381			.EN
382			.EQ I
383			bold rho sub s ~=~ left {~ lpile {{bold r sub s} above 0 } ~~~
384			lpile {{if~a sub 5^=^0~or~bold r sub s^>^t sub s} above otherwise }
385			.EN
386			.EQ I
387			bold rho sub a ~=~ left {~ lpile {{ bold p^bold C^
388			(1~-~bold r sub s ) (1~-~a sub 6 )} above
389			{bold p^bold C^(1~-~a sub 6 )}} ~~~ lpile
390			{{if~a sub 5^=^0~or~bold r sub s^>^t sub s} above otherwise }
391			.EN
392			.EQ I
393			bold tau sub d ~=~ a sub 6 (1~-~bold r sub s ) (1~-~a sub 7 )^bold p^bold C
394			.EN
395			.EQ I
396			bold tau sub si ~=~ left {~ lpile {{a sub 6 a sub 7 (1~-~bold r sub s )^
397			bold p^bold C {g sub s ( q hat sub i )} over
398	greg	1.3	{(- q hat sub i cdot n hat sub p ) cos sub 1}} above
399	greg	1.1	0 } ~~~ lpile { {if~a sub 5^>^0} above otherwise }
400			.EN
401			.EQ I
402			bold tau sub s ~=~ left {~ lpile {{a sub 6 a sub 7 (1~-~bold r sub s )^
403			bold p^bold C} above 0} ~~~ lpile
404			{{if~a sub 5^=^0~or~a sub 6 a sub 7 (1~-~bold r sub s )^>^t sub s}
405			above otherwise}
406			.EN
407			.EQ I
408			bold tau sub a ~=~ left {~ lpile {{a sub 6 (1~-~bold r sub s ) (1~-~a sub 7 )
409			^bold p^bold C} above {a sub 6 (1~-~bold r sub s )^bold p^bold C}} ~~~
410			lpile {{if~a sub 5^=^0~or~a sub 6 a sub 7 (1~-~bold r sub s )^>^t sub s}
411			above otherwise}
412			.EN
413			.EQ I
414			bold r sub s ~=~ a sub 4
415			.EN
416			.EQ I
417			f sub s ( q hat sub i ) ~=~ e sup{[ ( h vec sub i cdot n hat sub p ) sup 2
418			~-~ \|\| h vec \|\| sup 2 ]/
419			( h vec sub i cdot n hat sub p ) sup 2 / alpha sub i}
420			over {4 pi alpha sub i}
421			.EN
422			.EQ I
423			alpha sub i ~=~ a sub 5 sup 2 ~+~ omega sub i over {4 pi}
424			.EN
425			.EQ I
426			g sub s ( q hat sub i ) ~=~ e sup{( 2 q hat sub i cdot t hat~-~2)/ beta sub i}
427			over {pi beta sub i}
428			.EN
429			.EQ I
430			t hat ~=~ {v hat~-~d vec}over{\|\| v hat~-~d vec \|\|}
431			.EN
432			.EQ I
433			beta sub i ~=~ a sub 5 sup 2 ~+~ omega sub i over pi
434			.EN
435			.LP
436			The same caveat applies to specular rays generated for trans type as did
437			for plastic and metal.
438			Namely, if the roughness is greater than zero and the reflected ray,
439			$bold R ( P vec sub s ,~ r hat )$,
440			or the transmitted ray,
441			$bold R ( P vec sub s ,~ t hat )$,
442			intersects a light source, then it is not used in the calculation.
443			.NH 2
444			Anisotropic Types
445			.LP
446			The anisotropic reflectance types (plastic2, metal2, trans2) use the
447			same formulas as their counterparts with the exception of the
448			exponent terms, $f sub s ( q hat sub i )$ and
449			$g sub s ( q hat sub i )$.
450			These terms now use an additional vector, $b vec$, to
451			orient an elliptical highlight.
452			(Note also that the argument numbers for the type trans2 have been
453			changed so that $a sub 6$ is $a sub 7$ and $a sub 7$ is $a sub 8$.)\0
454			.EQ I
455			f sub s ( q hat sub i ) ~=~
456			1 over {4 pi sqrt {alpha sub ix alpha sub iy} }
457			exp left [ -^{{( h vec sub i cdot x hat )} sup 2 over{alpha sub ix}
458			~+~ {( h vec sub i cdot y hat )} sup 2 over{alpha sub iy}} over
459			{( h vec sub i cdot n hat sub p ) sup 2}right ]
460			.EN
461			.EQ I
462			x hat ~=~ y hat~times~n hat sub p
463			.EN
464			.EQ I
465			y hat ~=~ {n hat sub p~times~b vec}over{\|\| n hat sub p~times~b vec \|\|}
466			.EN
467			.EQ I
468			alpha sub ix ~=~ a sub 5 sup 2 ~+~ omega sub i over {4 pi}
469			.EN
470			.EQ I
471			alpha sub iy ~=~ a sub 6 sup 2 ~+~ omega sub i over {4 pi}
472			.EN
473			.EQ I
474			g sub s ( q hat sub i ) ~=~
475			1 over {pi sqrt {beta sub ix beta sub iy} }
476			exp left [ {{( c vec sub i cdot x hat )} sup 2 over{beta sub ix}
477			~+~ {( c vec sub i cdot y hat )} sup 2 over{beta sub iy}} over
478			{{( n hat sub p cdot c vec sub i )}sup 2 over
479			{\|\| c vec sub i \|\|}sup 2 ~-~ 1}
480			right ]
481			.EN
482			.EQ I
483			c vec sub i ~=~ q hat sub i~-~t hat
484			.EN
485			.EQ I
486			t hat ~=~ {v hat~-~d vec}over{\|\| v hat~-~d vec \|\|}
487			.EN
488			.EQ I
489			beta sub ix ~=~ a sub 5 sup 2 ~+~ omega sub i over pi
490			.EN
491			.EQ I
492			beta sub iy ~=~ a sub 6 sup 2 ~+~ omega sub i over pi
493			.EN
494			.NH 2
495			BRDF Types
496			.LP
497			The more general brdf types (plasfunc, plasdata, metfunc, metdata,
498			BRTDfunc) use the same basic formula given in equation (1),
499			but allow the user to specify $bold rho sub si$ and $bold tau sub si$ as
500			either functions or data, instead of using the default
501			Gaussian formulas.
502			Note that only the exponent terms, $f sub s ( q hat sub i )$ and
503			$g sub s ( q hat sub i )$ with the radicals in their denominators
504			are replaced, and not the coefficients preceding them.
505			It is very important that the user give properly normalized functions (ie.
506			functions that integrate to 1 over the hemisphere) to maintain correct
507			energy balance.