Coordinate Systems - Essentials of Interactive Computer Graphics - page 260

where

= x wc y wc ,

V wc

V ndc = V wc M w 2 n ,

V dc = V ndc M n 2 d ,

or

V dc = V ndc M n 2 d

= V wc M w 2 n M n 2 d .

If we let

M w 2 d = M w 2 n M n 2 d , (10.11)

then

V dc = V wc M w 2 d . (10.12)

Recall that M n 2 d is defined by Equation (10.6) and that M w 2 n is defined by Equa-
tion (10.10) (re-listing the two equations):

W dc
2

H dc
2 )

W dc
2

H dc
2 ) ,

M n 2 d = S ( , T ( , ( 10 . 6 )

2
W wc ,

2
H wc ) .

M w 2 n = T ( − cx wc ,− cy wc ) S ( ( 10 . 10 )

In this way,

Consecutive scaling opera-
tors. Two consecutive scaling
operators:

V dc = V wc M w 2 d

= V wc M w 2 n M n 2 d

S ( s x 1 , s y 1 ) S ( s x 2 , s y 2 )
have the same effect as scaling
once by the combined effect of
the scaling factors:

2
W wc ,

2
H wc )

W dc
2

H dc
2 )

W dc
2

H dc
2 ) ,

= V wc T ( − cx wc ,− cy wc ) S ( S ( , T ( ,

or

W dc
W wc ,

H dc
H wc )

W dc
2

H dc
2

S ( s x 1 s x 2 , s y 1 s y 2 ) .

V dc = V wc T ( − cx wc ,− cy wc ) S ( T ( , ) . (10.13)

Comparing Equations (10.12) and (10.13), we see that

W dc
W wc ,

H dc
H wc )

W dc
2

H dc
2

M w 2 d = T ( − cx wc ,− cy wc ) S ( T ( , ) . (10.14)

Equation (10.14) says that the operator M w 2 d , which transforms from the WC
(

x wc , y wc ) to the DC ( x dc , y dc ) , does the following.

• Move. The center of the WC window to the origin with ( T ( − cx wc ,− cy wc ) ).

H wc rectangle centered at the origin.
• Scale. With the WC window width of W wc , the scaling factor W dc

The result of this transform is a W wc ×

W wc changes
the width to W dc . In a similar fashion, the height becomes H dc .Afterthe
scaling operator, we have a W dc × H dc rectangle centered at the origin.