Joints from space occupancy
Hinton found that the ways two rectangles' edges cross are too varied to define a good knee (section 2.5.1). Instead each part, once it has a proximal and a distal end, is divided into zones: two ends, two halves and two poles. A joint is then a short list of zone pairs that must overlap and zone pairs that must not, plus a few rules on proportions (Fig. 2.13b).
The overlap test
Each zone is a convex polygon. Each of its borders is a line a·x + b·y = c, with the zone on its on-side, where a·x + b·y ≥ c. Points on a border count as on-side, so touching zones overlap. Appendix 1 shows that two convex polygons are disjoint exactly when one of them has a border with the other polygon wholly on its off-side. The tester checks every border of both zones and draws the one with the widest clearance, with its normal pointing from the on-side to the off-side.
Where the numbers come from
The zone fractions and joint tests are taken from the program listing in Appendix 3 (DEFTOPEND to DEFBOTPOLE, and KNEEJOIN, LEGJOIN, ARMJOIN, NECKJOIN, HEADJOIN, TERMJOIN). The ends are a fifth of a part's length, not a quarter. The poles are short central bars set back from each end, not the end caps. The elbow uses the knee test, and the ankle uses the wrist test.
A head's proximal end is the top of the head, because proximal means nearest the top of the head (section 2.5). So at the neck-head joint, the head's distal end meets the neck's proximal end.
The whole-overlap test is listed first because the program only tested joints between rectangles that already overlapped.