SECTION 9.5
833
3D Artwork
As described in Section 4.2, positions are defined in terms of pairs of
x
and
y
co-
ordinates on the Cartesian plane. The origin of the plane specifies the location (0,
0);
x
values increase to the right and
y
values increase upward. For three-dimen-
sional graphics, a third axis, the
z
axis, is required. The origin is therefore at (0, 0,
0); positive
z
values increase going into the page.
In two-dimensional graphics, the transformation matrix transforms the position,
size, and orientation of objects in a plane. It is a 3-by-3 matrix, where only six of
the elements can be changed; therefore, the matrix is expressed in PDF as an ar-
ray of six numbers:
a b
0
c d
0
=
a b c d tx ty
tx ty
1
In 3D graphics, a 4-by-4 matrix is used to transform the position, size, and orien-
tations of objects in a three-dimensional coordinate system. Only the first three
columns of the matrix can be changed; therefore, the matrix is expressed in PDF
as an array of 12 numbers:
a
d
g
tx
b
e
h
ty
c
f
i
tz
0
0
=
a b c d e f g h i tx ty tz
0
1
3D coordinate transformations are expressed as matrix transformations:
a
d
x' y' z'
1
=
x y z
1
×
g
tx
b
e
h
ty
c
f
i
tz
0
0
0
1
Carrying out the multiplication has the following results:
x'
=
a
×
x
+
d
×
y
+
g
×
z
+
tx
y'
=
b
×
x
+
e
×
y
+
h
×
z
+
ty
z'
=
c
×
x
+
f
×
y
+
i
×
z
+
tz
Position and orientation of 3D artwork typically involves translation (movement)
and rotation along any axis. The virtual camera represents the view of the art-