Perspective Projection - Transformations and Projections in Computer Graphics - page 115

As a simple application of these results, let's select h = v =45 ◦ , which implies
m = n = 1. Let's also assume screen dimensions of 100 100 pixels, a local origin at

the center of the screen, and k =1. Forpoint P =(1 , 2 , 1), we get the scale factors

×

m ( z + k ) = −

− x 1

1+1 =

y
n ( z + k ) =

2
1+1 =1 .

c x = − 0 . 5 , y =

Thus, the screen coordinates are s x =50 50 = 50 (the

top of the screen). However, any point with coordinates (1 ,y, 1) where y> 2would
produce a scale factor c y > 1, implying that its projection is outside the screen.

× ( − 0 . 5) = − 25 and s y =1 ×

Exercise 3.27: Why is Equation (3.18) asymmetric with respect to x and y (i.e., why
−

x and not − y )?

3.9.1 Perspective Depth
The perspective projection converts a three-dimensional point to a two-dimensional
point. It completely erases any information about the depth (the z coordinate) of the
original point. However, certain algorithms for hidden surface removal need precisely
such information. We therefore need to generalize our perspective projection to create
a third coordinate z ∗ with information about the original z coordinate of the projected
point. The obvious choice is z ∗

= z , but this has a serious downside: It does not

preserve straight lines.
Imagine two three-dimensional points, P 1 =( x 1 ,y 1 ,z 1 )and P 2 =( x 2 ,y 2 ,z 2 ),
projected to the points

P 1 = x 1 k

k + z 1 ,z 1

P 2 = x 2 k

k + z 2 ,z 2 .

y 1 k y 2 k

k + z 1 ,

and

k + z 2 ,

Note that the two projected points are not necessarily on the projection plane. We say
that they are located in the image space .
The straight segment P ( t )= P 1 +( P 2 − P 1 ) t (Equation (Ans.7)) connects the two

original points, while the segment P ∗ ( u )= P 1 +( P 2 − P 1 ) u connects the two projected

ones. It can be shown that an arbitrary point P ( t 0 )on P ( t ) is projected to a point
that's not on P ∗ ( u ).
This is why the perspective depth projection is not chosen simply as z ∗ = z but as
z ∗ = z/ ( k + z ). This definition preserves depth information, because it has the property
z 1 >z 2 ⇒

z 1 >z 2 . It also preserves straight lines.

Exercise 3.28: Prove the claim above.