Pro OpenGL ES for iOS

46 CHAPTER 2: All That Math Jazz^

With these boundaries established, the one final transformation is that to the viewport,

OpenGL’s version of your screen. This is the point where OpenGL is fed the screen’s

dimensions, those of your display area, and the origin, which is likely the lower-left

corner of the screen. On small devices such as the iPhone or iPad, you will likely fill up

the entire screen and so will use the screen’s width. But should you want to place the

image into a subwindow of the main display, you could simply pass smaller values to the

viewport. The law of similar triangles plays out here.

In Figure 2-11 we want to find what the projected x’ is, given the x of an arbitrary vertex

on the model. Consider two triangles, one formed by the corners CBA and the other

smaller one by COA’ ( the O is for origin). The distance from C (where the eye is, to O is

d). The distance from C to B is d+z. So, just taking the ratio of those, as follows:

eye z deye

x

d

x

+

=

′

and

eye z deye

y

d

y

+

=

'

yields the following:

eye

eye

z d

xd

x

+

'= and

eye

eye

z d

yd

y

+

'= =

Figure 2-11. Mapping a vertex to the viewport using the Law of Similar Triangles

Figure 2-12 shows the final translations. Those can be added to x’ and y’:

x

eye

eye

T

z d

xd

x +

+

'= and y

eye

eye

T

z d

yd

y +

+

'=