Transformations - Transformations and Projections in Computer Graphics - page 17

y

j

θ

x

θ

i

Figure 1.8: Scaling Along Rotated Axes.

Exercise 1.24: Discuss scaling relative to a point ( x 0 ,y 0 ), and show that the result
is identical to the product of a translation followed by scaling, followed by a reverse
translation.
Using Equation (Ans.2) in the Answers to Exercises, it is easy to explore the effect
of two consecutive scaling transformations, with scaling factors of k 1 and k 2 and about
points P 1 =( x 1 ,y 1 )and P 2 =( x 2 ,y 2 ), respectively. We simply multiply the two

transformation matrices
⎛ ⎞ ⎛ ⎞

k 1 0 0 k 2 0 0

⎝ ⎠ ⎝ ⎠

0 k 1 0 0 k 2 0

x 1 (1 − k 1 ) y 1 (1 − k 1 )1 x 2 (1 − k 2 ) y 2 (1 − k 2 )1

⎛ ⎞

k 1 k 2 0 0

⎝ ⎠ .

= 0 k 1 k 2 0 (1.16)

k 2 (1 − k 1 ) x 1 +(1 − k 2 ) x 2 k 2 (1 − k 1 ) y 1 +(1 − k 2 ) y 2 1

The result is similar to Equation (Ans.2) except for the bottom row. It seems that the
product of two scalings is a third scaling with a factor k 1 k 2 , but about what point? To
write Equation (1.16) in the form of Equation (Ans.2), we write

k 2 (1 − k 1 ) x 1 +(1 − k 2 ) x 2 = x c (1 − k 1 k 2 ) ,

k 2 (1 − k 1 ) y 1 +(1 − k 2 ) y 2 = y c (1 − k 1 k 2 ) ,

and solve for ( x c ,y c ), obtaining

x c = k 2 (1

− k 1 ) x 1 +(1 − k 2 ) x 2

,

1 − k 1 k 2

y c = k 2 (1

− k 1 ) y 1 +(1 − k 2 ) y 2

.

1 − k 1 k 2

The center of the double scaling is therefore point

P c = k 2 (1

− k 1 ) 1 − k 2

P 1 +

k 1 k 2 P 2 = a P 1 + b P 2 .

1 − k 1 k 2 1 −