Graphics Reference
In-Depth Information
We add to this the idea that often, writing such a program removes the neces-
sity of understanding the details in the future: If you've done a good job, you can
reuse your program.
We'll use conventional mathematical notation, in which most variables appear in
italics; vectors will be written in roman boldface (e.g.,
u
), as will matrices. In
general, vectors will be lowercase, matrices uppercase. When a variable has a
subscript used for indexing, it's italic, as in the
i
in
i
x
i
. When a subscript or
superscript is mnemomic, as in
ρ
dh
, the “directional hemispherical reflectance,”
it's in roman font.
Certain special sets have predefined names and are written in boldface font:
R
is the set of real numbers;
C
is the set of complex numbers;
R
+
(pronounced
“R plus”) is the set of positive real numbers; and
R
0
(pronounced “R plus zero”)
is the set of non-negative reals.
Sets are generally denoted by capital letters. The
Cartesian product
of the sets
B
and
C
is the set
1
,
B
×
C
=
{
(
b
,
c
):
b
∈
B
,
c
∈
C
}
(7.1)
which is pronounced “B cross C”; despite this, it is called a “Cartesian” product
rather than a “cross product”; that term is reserved for the cross product of vectors
discussed in Section 7.6.4.
The product
R
R
is denoted
R
2
; higher order products are
R
3
,
R
4
, etc., with
the
n
-fold product being
R
n
.
The
closed interval
[
a
,
b
]
is the set of all real numbers between
a
and
b
, inclu-
sive, that is,
×
[
a
,
b
]=
{
x
:
a
≤
≤
}
x
b
.
(7.2)
If
b
a
, then the interval is empty; if
b
=
a
, the interval contains just the num-
ber
b
. We'll also occasionally use intervals that contain just one of their endpoints
(i.e.,
half-open intervals
):
<
[
a
,
b
)=
{
x
:
a
≤
x
<
b
}
,
(7.3)
(
a
,
b
]=
{
x
:
a
<
x
≤
b
}
.
(7.4)
We also define the following two notational conventions:
[
a
,
∞
)=
{
x
:
a
≤
x
}
,
(7.5)
(
−∞
,
b
]=
{
x
:
x
≤
b
}
.
(7.6)
1. This notation means “the set of all pairs (
b
,
c
) such that
b
is in
B
and
c
is in
C
.” That is,
the colon is read “such that.”
