Where does the matrix P come from? Is it just to force the homogeneous coordinates to give us the desired result?
Why do we throw out the third component after the homogeneous divide?
@merc You can certainly think of it this way: as just a way to implement perspective projection, using the standard 4x4 matrices and homogeneous interpretation of vectors that's available in the standard rasterization pipeline. If you want a deeper view of the relationship between perspective projection and homogeneous coordinates, take a look at projective geometry.
@cupoftea Because it's always 1, and hence doesn't give you any useful information. ;-). In other words, you just want the projected 2D x and y coordinates. In reality you'll also need a depth value, in order to handle occlusion; if you want a bit more realistic discussion of what happens in the rasterization pipeline, check out this link. We'll talk about some (but not all) of this in a later lecture.
Is there any specific reason we need a 4x4 matrix for this, or is it just because that's the system we're working in? Because it seems like the 4th term in the vectors don't do anything.