Perspective projections are often preferred because they make the more distant objects appear smaller than those closer to the viewpoint. They involve more calculation than parallel projections, but are often preferred for their greater realism. Note that a parallel projection may be considered as a special case of the perspective projection where the viewpoint is at infinity.

A perspective transformation produces a projection from a viewpoint E onto a given plane. Because you can always move the axes to ensure that the plane coincides with z=0 and the normal from the plane through the point E lies along the z-axis, you may restrict the discussion to this simple case.

Perspective Projection

The above figure shows an example of the perspective projection from the point E at (0,0,-d) to the z=0 plane.

The projection is obtained by joining E to each vertex in turn and finding the intersection of this line with the plane z=0. The vertices are then joined by straight lines to give the wire-frame drawing of the object in the plane.

This method of drawing the object, makes use of some of the well-known properties of perspective projections, namely that straight lines are projected into straight lines and facets ( A facet is a closed sequence of co-planar line segments, a polygon in other words) are projected into facets. Parallel sets of lines may be projected into a set of parallel lines or into a set of lines meeting at the `vanishing point'.

We may consider the equation of the projection either as the result of the transformation matrix or derive it from the following diagram

Consider the diagram first. This shows the y=0 plane with the Ox and Oz axes. The point of projection is E at the point (0,0,-d) on the z-axis, so the distance OE is of length d. The point P (with values x and -z) projects into the point P' while the point Q (with values X and Z) projects into the point Q'.

From the first set of similar triangles, we can see that d/x' = (d-z)/x and so x'=d*x/(d-z)

From the second set of similar triangles, we can see that d/X'=(d+Z)/x and so X'=d*X/(d+Z)

Thus if we are careful to take the correct sign for z in each case, we can quote the general rule:

x' = d*x/(d+z)

and we have a similar position for the y-coordinate when looking at the x=0 plane.

Now let us turn to the transformation matrix. In this case it becomes, 
	X		1	0	0	0		x
	Y	=	0	1	0	0		y
	Z		0	0	0	0		z	
	H		0	0	1/d	1		1
which gives the four equations.

	X = x		Y = y 		Z = 0		H = (z+d)/d

To get back to the homogeneous coordinates, we need to make H=1 and so 
we have to divide throughout by (z+d)/d. This gives:

		 X = d*x/(z+d), 	Y = d*y/(z+d),	Z=0   and   H=1 

Hence we get the same expression for this derivation.

The closer the point of projection, E, is to the object, the more widely divergent are the lines from E to the vertices and the greater the change in size of the projected object. Conversely, the further away we move E, the closer the lines get to a parallel set and the smaller the change in size of the object. Thus we may think of the parallel projection as being an extreme case of perspective when the point of projection E is an infinite distance from both the object and the plane.

This perspective projection is an example of a `single-point perspective' and the consequence of this is shown in the next figure.

The one set of parallel lines forming edges of the cube meet at the vanishing point, while the other sets meet at infinity (i.e. they remain parallel). The transformation matrix for this projection may be written in the form given below, where r = 1/d.

	1 	0 	0 	0 
T1 = 	0 	1 	0 	0 
	0 	0 	0 	0 
	0 	0 	r 	1

When we come to deal with two or three point perspectives, then we have two or three sets of parallel lines meeting at their respective vanishing points. The matrices for these are given below:

	1 	0 	0 	0 		1	0	0	0
T2 = 	0 	1 	0 	0 	T3 = 	0 	1	0	0
	0 	0 	0 	0 		0	0	0	0
	0 	q 	r 	1		p	q	r	1

The following figure shows an example of a three-point perspective.

We now have enough information to specify the form of a general transformation matrix.

		t{11} 	t{12} 	t{13} 	t{14}
	T = 	t{21} 	t{22} 	t{23} 	t{24}
		t{31} 	t{32} 	t{33} 	t{34}
		t{41} 	t{42} 	t{43} 	t{44}

This divides into four areas, each of which relates to a different form of transformation.

			T1	      	|   T3
	T = 	shear, scale & 	      	|  shift
		   rotate	      	|
		  ________________________________
		   T2 usually zero   	|  T4=1		

T2 is zero for all affine transformations and when we are dealing with perspective projections, the number of non-zero elements in T2 will tell us whether it is a one-, two- or three-point perspective.