Transformation Operators - Essentials of Interactive Computer Graphics - page 188

and the results of the resizing would, as expected, be V cs :
V cs = x cs y cs
=

dx i =2 dy i =3

V c S ( 3 , 0 . 5 )

dx o =6

Input
Points

= x c × s y

dy o =1.5

s x y c ×

= 1 5

× 35 × 0 .

= 32 5 . . Output

Points

Figure 8.7 shows that if we examine the relative distances between the two input

points (i.e., V a and V c ), we see that the distance in the x -direction is dx i :

Figure 8.7. Relative dis-
tances between input and
output points.

= | − | = | − | = |− | = ,

dx i x c x a 1 3 2 2

whereas the distance in the y -direction is dy i :

dy i = | y c − y a | = | 5 − 8 | = |− 3 | = 3 .

When we compare these distances to those of the output points (i.e., V at and V ct ),
we observe the x and y direction distances to be dx o and dy o :

dx o = | x cs − x as | = | 3 − 9 | = |− 6 | = 6 ,

dy o = | y cs − y as | = | 2 . 5 − 4 | = |− 1 . 5 | = 1 . 5 .

We observe that

dx o = s x × dx i = 3 × 2 = 6 ,

.
We see that when we apply the same scale operator to V a and V c , in effect we
are also applying the scale operator on the distances between these points. In this

dy o = s y × dy i = 0 . 5 × 3 = 1 . 5

Scaling operator. Changes
the x and y size of objects ac-
cording to the scale factors.

way, by repositioning each individual point, we have resized the relative distances
between these points. Figure 8.8 shows applying the S

( 3 , 0 . 5 ) operator on four

vertices that form a rectangle:

⎧
⎨

V a =( 3 , 8 ) ,

V b =( 1 , 8 ) ,

Input points :

⎩

V c =( 1 , 5 ) ,

V d =( 3 , 5 ) .

We see that the scale operator will reposition the input points to

V as = V a S ( 3 , 0 . 5 ) ,

V bs =

( , . ) ,

V b S 3 0 5

V cs = V c S ( 3 , 0 . 5 ) ,

V ds = V d S ( 3 , 0 . 5 ) ,