Get dividing out the also called the reciprocal divide

222 Chapter 5 Viewing and Projection

By similar triangles (both have the same angles), we get

Solving for xndc:

So our final projection transformation equations are

Recall that to transform from RP3to R3we need to divide the other coordi-nates by the w value. If we can set up our matrix to map −zv to our w value, we can take advantage of the homogeneous divide to handle the nonlinear part of our transformation. We can write the situation before the homogeneous divide as a series of linear equations:

x′=d a x
y′= dy
z′= dz
w′= −z

Let’s see how this matrix works in practice. If we multiply it by a generic point in view space, we get