Good starting point for another thread, John!
The happo-noh-kuzushi is a core tenet of judo, but it applies to everything.
The classic example is the line drawn through the feet. No matter how you stand, if you draw a line through your feet that is your strong line. It is very difficult to throw you in that direction.
Draw a line at 90 degrees to your strong line, bisecting it, and that is your weak line. The closer the forces in a throw are applied to the weak line, the easier it is to throw you. (AAAgh. I have all these neat diagrams and no way to show them.)
Since lines go both ways, each stance gives two good directions to throw. Natural stance- hard to throw side to side, easy front to back. Back stance: hard front or back, easy to either side. Zenkutsu dachi: weak forward left and right rear directions in a right lead stance.
The "8 points" are front, back, left, right and each of the "corners" between.
Experimentally, if you miss a sweep most often it is because the foot was swept along the strong line or the force applied to the upper body was along the strong line. Being a few degrees off can make a huge difference.
That was the classic explanation. Here's my patented, get-it-though-to-cynical-cops explanation:
Point #1:
No matter how you stand, your feet make a box. Outside edges of the foot make two of the lines and one line heel to heel and one line toe to toe. A box. (Draw the foot pattern to all of your stances and draw the box. Some, like a back stance are more of a triangle.)
Point #2: Your center of gravity is located about a hand width below your belly button and half way between your belt uckle and your spine. That's your dot.
If your dot ever leaves the box you will fall over and there's not a damn thing you can do about it. Even a limbo dancer...her COG stays directly over her feet.
A large part of throwing is figuring out what edge of the box the dot is closest to. Combine that with any momentum the threat is delivering and your throw is mostly done for you.
This explanation also works on the ground. If someone is pinning you, all their points of contact with the ground- hips, feet elbow, head, hands constitute their box. Their COG remains the same. It is easier to roll them to the nearest edge of the box to escape.
I'm sure this must be crystal clear. Really.
Rory