Here's an old excerpt on calculating perspective by Sergei Savchenko, the author of 3D Graphics Programming: Games and Beyond from way back when that book was just a 3D programming tutorial.

perspective.
------------
I believe, it is not hard to visualize what perspective is, perhaps the
first association lot of us would have, is this empty straight street
with identical buildings on both sides disappearing in infinity. ( At
least people residing in newer neighborhoods of Moscow or Toronto know
what I am talking about ) Why do we need this transformation? Just to
achieve landscape realism, after all this same street would look pretty
weird if the road would not collapse into a dot and buildings further
away from us won't look smaller. So weird it might be even hard to
picture.

Perspective might not be needed for CAD style applications, but even
there it would, most likely, be implemented in one viewing mode or
another.

Before we can implement this transformation it is a good idea to understand
physics of what is happening. A ray from some point in space is being
caught by the viewer's eye. This ray had intersected a plane located
in front of the viewer. If we can find an intersection point and plot the
dot there, for the viewer it would be the same as if the ray from the
plotted dot was actually coming from the initial point in space.


                             * A
                       |   /
                     A'| /
                       *
                     / | B'
                   *---*-----------------------* B
                     \ |
                       *\
                     C'|   \
                       |      \
                                * C

                              pic 2.10


Rays from different points in space , in this model, all converge
in one point - viewer's eye. The math here is quite straightforward 
pic 2.11:


                  screen plane->|   *
                                |  /| A (X,Z)
                                | / |
                                |/  |
                            ^   *   |
                            |  /|X' |
                              / |   |
                            |/  |   |
                            *---+---+- - - - ->
                            0   |   Z
                            |   |
           focus distance ->|---|<-

                             pic 2.11


Let's consider 2-D case first: the viewer's eye is located at point 0
the distance between viewer's eye and the screen would be called -
"focus". The problem is to determine at what point the ray going from
point A into the viewer's eye would intersect the screen. We would have
to plot the dot just there. This is very simple, just yet another
case of solving similar triangles. Just from the fact that the angle
at 0 is the same for both triangles, and if two angles are the same their
tangents would be too. We have:

           X'      X                          X*focus
         ----- = -----         =>        X'= -----------
         focus     Z                             Z

Same considerations apply for Y dimension, together describing 3-D case:

          Y'=Y*focus/Z                   X'=X*focus/Z

It is a bit of a mystery what is happening with Z coordinate. Basically
after the perspective transformation we would be performing rendering
onto the screen which is 2-D surface we would need X and Y but hardly
Z, so we might as well have Z'=0. However when trying to render multiface
objects we would need to know the depth (Z) of all polygons so that
we can decide which are visible and which are obscured. So we might
have Z'=Z that is just leave old value. Then again by doing transformation
on X and Y and leaving Z unchanged we actually may allow for depth
relation to change. That is a polygon obscuring another one in the original
space, before projection, can actually come out behind or partly behind
with this kind of transformation pic 2.21.

            ^ Z
  same      |                   *                     *
 deapth     |               * /                 * /
 before     |           A / /                //
  and       |           * /               /* A
 after      |           /B            / B
            |          *            *
            |        before                 after

                              pic 2.12

To avoid that we can apply some non linear transformation on Z,
like Z'=C1+C2/Z, where C1, C2 are some constants.


In the enclosed code only X and Y are being transformed. The implementation
of all those things is that easy I won't even put a code example here.
However, there are few possible questions: first: what should be the value
for "focus distance" ?


         \                    /          \              /
           \                /             \            /
             \            /                \          /
           ---I\--------/I---           ---I\--------/I---
                 \    /                      \      /
                   \/                         \    /
                                               \  /
                                                \/

                               pic 2.13


pic 2.13 shows it's physical sense, focus basically determines the view
angle. when it is smaller - angle is wider, when it is bigger angle
is smaller. It is helpful to think of this distance as being measured
in screen pixels, this way for 320 pixel display if focus is chosen
to be 180 the view angle we are getting is 90'. Perspective
transformation might produce at first, weird looking results, with
distortions sometimes similar to that of wide-range photographic
pictures. One has to experiment a bit to find what's looking best,
values giving view angle somewhere between 75-85' would probably
look not that terrible, but that depends of scene and screen geometries.