Frames of Reference in the Plane - Rotation Transforms for Computer Graphics - page 105

Fig. 8.1 The X Y axial
system is coincident with XY

Fig. 8.2 The X Y axial
system is translated (t x ,t y )

Figure 8.2 shows the frame X Y translated by (t x ,t y ) which is equivalent to
translating P by T − t x , − t y which is also the inverse transform T − 1

t x ,t y . Therefore,

a point P(x,y) in XY has coordinates P (x ,y ) in X Y given by
⎡

⎤ ⎡ ⎤

x
y
1

x
y
1

⎣ ⎦ =

T − 1
t x ,t y

⎣ ⎦

where

⎡ ⎤

10 − t x

T − 1

⎣ ⎦ .

t x ,t y =

01 − t y

00 1

For example, the point (t x ,t y ) in XY should have coordinates ( 0 , 0 ) in X Y :
⎡ ⎤ ⎡ ⎤ ⎡ ⎤

0
0
1

10 − t x t x

t y
1

⎣ ⎦ = ⎣ ⎦ ⎣ ⎦

01 − t y

00 1

which is confirmed.