Frames of Reference in the Plane - Rotation Transforms for Computer Graphics - page 106

Fig. 8.3 The X Y axial
system is rotated β

8.3.2 Rotated Frame of Reference

Figure 8.3 shows the frame X Y rotated β which is equivalent to rotating P by
− β and is effected by the transform R − 1
β

. Therefore, a point P(x,y) in XY has

coordinates P (x ,y ) in X Y given by
⎡

⎤ ⎡ ⎤

x
y 1

x
y
1

⎣ ⎦ =

R − 1
β

⎣ ⎦

where

⎡ ⎤

cos β sin β 0

R − 1
β

⎣ ⎦ .

= − sin β cos β 0

0 0 1

We can also confirm this using the geometry shown in Fig. 8.3 ,

x = R cos θ

=

R sin θ
x = R cos (θ − β)
y =

y

R sin (θ − β)

x = R sin θ sin β

= x cos β + y sin β
y =

R cos θ cos β +

R sin θ cos β − R cos θ sin β

=− x sin β + y cos β

which as a homogeneous matrix is
⎡ ⎤ ⎡ ⎤ ⎡

⎤
⎦

x
y 1

cos β sin β 0 x

y
1

⎣ ⎦ = ⎣ ⎦ ⎣

− sin β cos β 0

0 0 1