Perspective Projection - Transformations and Projections in Computer Graphics - page 89

Exercise 3.8: Show why point (0 , 1 / 4) is not located on P ∗ ( t ).
Transforming and projecting . This example illustrates the advantage of the
projection matrix T p of Equation (3.4). Given an object, we might want to transform
it before we project its points. In such a case, all we have to do is prepare the individual
4 4 transformation matrices, multiply them together in the order of the transforma-

tions, and multiply the result by T p . Assume that we want to apply the following
transformations to our object: (1) Rotate it about the x axis by 90 ◦ from the direction
of positive y to the direction of positive z (Figure 3.25a). (2) Translate it by 3 units in
the positive z direction. (3) Scale it by a factor of 1/2 (i.e., shrink it to half its size) in
the y dimension. The three transformation matrices are

×

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

1000
0010
0 − 100
0001

1000
0100
0010
0031

1000
01 / 200
0010
0001

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

T R = , T T = , T S =

and their product with T p (we assume k =1,so r = 1) produces

⎡ ⎤ ⎡ ⎤

1000
0100
0001
0001

1000
0001
0

⎣ ⎦ ⎣ ⎦

T = T R T T T S = . (3.5)

1 / 200
0004

−

y

y

z

z

x

x

(a) (b)

Figure 3.25: Rotation about the x Axis.

We can now pick any point on the object, write it as a 4-tuple in homogeneous
coordinates, and multiply it by T to obtain its projection after applying the three
transformations to it. Notice that a point cannot be scaled, but the effect of scaling is
to move points such that the scaled object will shrink to half its size in the y dimension.
As an example, multiplying point (0 , 1 , 4 , 1) by T results in (0 , 2 , 0 , 5),which,after

dividing by the fourth coordinate, produces the two-dimensional point (0 , 2 / 5).

−

Exercise 3.9: Multiply point (0 , 1 , − 4 , 1) by the product T R T T T S and explain the

result.