Frames of Reference in the Plane - Rotation Transforms for Computer Graphics - page 109

Fig. 8.6 If the X -and
Y -axes are assumed to be
unit vectors, their direction
cosines form the elements of
the rotation matrix

8.4 Direction Cosines

Direction cosines are the cosines of the angles between a vector and the Cartesian
axes, and for a unit vector they are its components.
Figure 8.6 shows the rotated frame X Y , and by inspection the direction cosines
for a vector lying on X are cos β and cos ( 90° β) , which can be rewritten as cos β

and sin β . The direction cosines for a vector lying on Y are 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 inverse rotation matrix R − 1
β

− :

.

cos β sin β

R − 1
β

=

− sin β cos β

The top row contains the direction cosines for the X -axis and the bottom row con-
tains the direction cosines for the Y -axis. This relationship also holds in 3D. Conse-
quently, if we have access to these cosines we can construct a transform that relates
rotated frames of reference.
Figure 8.7 shows an axial system X Y rotated 45°, and the associated transform

is

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

x
y
1

0 . 707 0 . 707 0 x

y
1

⎣ ⎦ ≈ ⎣ ⎦ ⎣ ⎦ .

− 0 . 707 0 . 707 0

0 0 1

Fig. 8.7 The four vertices of
the unit square shown in both
frames