Transformation - Mathematics for Computer Graphics - page 57

x -coordinate. And finally, we add 1 to the reflected coordinate to compensate
for the original subtraction. Algebraically, the three steps are

x 1 = x − 1

x 2 = − ( x − 1)

x = − ( x − 1) + 1

which simplifies to

x =

− x +2

y = y

(7.30)

or in matrix form,

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

x
y
1

− 102

010
001

x
y
1

⎣ ⎦ = ⎣ ⎦ · ⎣ ⎦

(7.31)

Figure 7.5 illustrates this process.
In general, to reflect a shape about an arbitrary y -axis, y = a x , the fol-
lowing transform is required:
x = − ( x − a x )+ a x = − x +2 a x

y = y

(7.32)

or, in matrix form,

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

x
y
1

− 10 a x

01 0
00 1

x
y
1

⎣ ⎦ = ⎣ ⎦ · ⎣ ⎦

(7.33)

Y

X

− 2 − 1 0 1 2 3 4

Fig. 7.5. The shape on the right is reflected about the x = 1 axis.