Coordinate Systems - Essentials of Interactive Computer Graphics - page 259

10.4 The World Coordinate Window

In order to properly support WC design space, in our programs we must identify
a rectangular region of interest, or the window inside the WC:

⎧
⎨

center =( cx wc , cy wc ) ,

WC window = width = W wc ,

⎩

height = H wc .

We must then construct a corresponding M w 2 n to transform this region to the
NDC space. In general, we transform the center of the region to the origin

( T ( − cx wc ,− cy wc ) ) and then scale the width and height of the region to the NDC

2 2

2 × 2 square ( S (

W wc , H wc )

):

2
W wc ,

2
H wc ) .

M w 2 n = T ( − cx wc ,− cy wc ) S ( (10.10)

10.4.1 Vertices in Different Coordinate Spaces

As programmers working with graphics APIs, we specify vertices in the WC, or
V wc . As illustrated in Figure 10.16, this vertex undergoes different transforms
until it reaches the DC, or V dc , before being displayed on the output drawing
area:

M w 2 n
−→

M n 2 d
−→ V wc V ndc V dc ,

V wc =(x wc, y wc ) V ndc =(x ndc, y ndc ) V dc =(x dc, y dc )

WC
W indow

M n2d

M w2n

1

H dc

H wc

2

(cx wc, cy wc )

1

-1

-1

W wc

W dc

2

WC NDC DC

Figure 10.16. Transformation of a vertex between different coordinate systems.