The Nitty Gritty Details

If we let k = n^T l and cancel terms we find we have the matrix equation

x′  y′  z′  w′

=

(k−alx)  −blx  −clx  −dlx  −aly  (k−bly)  −cly  −dly  −alz  −blz  (k−clz)  −dlz  −alw  −blw  −clw  (k−dlw)

sx  sy  sz  1

It's not difficult to see that w' = n^Tl−n^Tsl_w. For both possible values of l_w (1 or 0), we find that w' = β so the point p' is transformed into the point p by perspective division.