Graphics Reference
In-Depth Information
P
(
x
,
y
)
Y
x
P'
(
x'
,
y'
)
Y'
y'
X'
x'
y
b
X
Fig. 7.16.
The secondary set of axes are rotated by
β
.
When a coordinate system is rotated and translated relative to the refer-
ence system, a point
P
(
x
,
y
) has coordinates (
x
,y
) relative to the new axes
given by
⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
x
y
1
cos(
β
) in
β
)0
10
−
t
x
x
y
1
⎣
⎦
=
⎣
⎦
·
⎣
⎦
·
⎣
⎦
−
sin(
β
)
β
)0
0
01
t
y
00 1
−
0
1
which simplifies to
⎡
⎤
⎡
⎤
⎡
⎤
x
y
1
cos(
β
) in
β
)
−
t
x
cos(
β
)
−
t
y
sin(
β
)
x
y
1
⎣
⎦
=
⎣
⎦
·
⎣
⎦
−
sin(
β
)
β
)
t
x
sin(
β
)
−
t
y
cos(
β
)
(7.74)
0
0
1
7.6 Direction Cosines
Direction cosines are the cosines of the angles between a vector and the axes,
and for unit vectors they are the vector's components. Figure 7.17 shows two
unit vectors
X
and
Y
, and by inspection the direction cosines for
X
are
cos(
β
) and cos(90
◦
−
β
), which can be rewritten as cos(
β
) and sin(
β
), and the
direction cosines for
Y
cos(90
◦
+
β
) and cos(
β
), which can be rewritten as
−
sin(
β
) and cos(
β
). But these direction cosines cos(
β
)
,
sin(
β
)
,
−
sin(
β
)and
cos(
β
) are the four elements of the rotation matrix used above:
cos(
β
) in
β
)
(7.75)
−
sin(
β
)
β
)
The top row contains the direction cosines for the
X
-axis and the bottom
row contains the direction cosines for the
Y
-axis. This relationship also holds
in 3D.
