Graphics Reference
In-Depth Information
which in matrix form is
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
·
⎡
⎣
⎤
⎦
x
y
z
1
s
x
000
0
s
y
00
00
s
z
0
0001
x
y
z
1
=
(7.51)
The scaling is relative to the origin, but we can arrange for it to be relative
to an arbitrary point (
p
x
,p
y
,p
z
) with the following algebra:
x
=
s
x
(
x
−
p
x
)+
p
x
y
=
s
y
(
y
−
p
y
)+
p
y
z
=
s
z
(
z
−
p
z
)+
p
z
(7.52)
which in matrix form is
⎡
⎤
⎡
⎤
⎡
⎤
x
y
z
1
s
x
s
x
)
0
s
y
0
p
y
(1
− s
y
)
00
s
z
p
z
(1
− s
z
)
000
00
p
x
(1
−
x
y
z
1
⎣
⎦
⎣
⎦
·
⎣
⎦
=
(7.53)
1
7.4.3 3D Rotations
In two dimensions a shape is rotated about a point, whether it be the origin
or some arbitrary position. In three dimensions an object is rotated about an
axis, whether it be the
x
-,
y
-or
z
-axis, or some arbitrary axis. To begin with,
let's look at rotating a vertex about one of the three orthogonal axes; such
rotations are called
Euler rotations
after the Swiss mathematician Leonhard
Euler (1707-1783).
Recall that a general 2D-rotation transform is given by
⎡
⎤
⎡
⎤
⎡
⎤
x
y
1
cos(
β
)
−
sin(
β
)0
x
y
1
⎣
⎦
=
⎣
⎦
·
⎣
⎦
sin(
β
)
cos(
β
)0
(7.54)
0
0
1
which in 3D can be visualized as rotating a point
P
(
x, y, z
) on a plane parallel
with the
xy
-plane as shown in Figure 7.8. In algebraic terms this can be written
as
x
=
x
cos(
β
)
y
sin(
β
)
y
=
x
sin(
β
)+
y
cos(
β
)
z
=
z
−
(7.55)
Therefore, the 3D transform can be written as
⎡
⎤
⎡
⎤
⎡
⎤
x
y
z
1
cos(
β
)
−
sin(
β
)00
x
y
z
1
⎣
⎦
⎣
⎦
·
⎣
⎦
sin(
β
)
cos(
β
)00
=
(7.56)
0
0
1
0
0
0
0
1
