ray/doc/materials.1

.\" RCSid "$Id$"
.\"Print with tbl and eqn or neqn with -ms macro package
.vs 12
.nr VS 12
.nr PD .5v      \" paragraph distance (inter-paragraph spacing)
.EQ L
delim $$
.EN
.DA
.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
#	Content
1	.\" RCSid "$Id$"
2	.\"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	{( q hat sub i cdot n hat sub p ) cos sub 1}} above
322	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	{( q hat sub i cdot n hat sub p ) cos sub 1}} above
380	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	{(- q hat sub i cdot n hat sub p ) cos sub 1}} above
399	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.