I suppose it might be useful for beginners to offer them a framework that has a relatively limited set of angles, but I agree with you that in the long run strict adherence to such a technically flawed system can only block further improvement. I think that most players who start with a "limited-set fractional-ball aiming" system will not really see the problem because they will soon transition to playing the shots by feel. Or they won't be able to make many shots.Patrick Johnson said:Actually, I said it's a fantasy to think "there are only x shots", meaning I think it's a fantasy to think the number of shots can be quantified, much less limited to a few. ...
But for those of us who like to do technical analysis of this kind of thing, there is a more or less reasonable answer to "how many different cut angles are there?" I would phrase it a little differently. For the range of shots that you expect to make, how many different cut angles would be enough if you limited yourself to a finite set? A reasonable lower limit to this can be found with only minor hand-waving:
I expect to be able to make a ball off the foot spot into the head pockets. I hope to do this most of the time for cut angles less than 45 degrees. It is easy to figure out the angular size of the target and simply divide that into 45 or 90 degrees if you count cuts to both the left and right. What is the angular size of the target? I think that any aiming system needs to be able to pick out the middle half of a pocket for the target. A system that has a whole pocket's-width of uncertainty is not worth considering. From geometry it's easy to figure out that from the foot spot, the target at the head spot is only 1 degree wide (depending some on how large the pocket is).
This means that if you want to be able to shoot a shot that is 6 diamonds long consistently, your method of aiming must accommodate at least 45 (or 90 if you you count both left and right cuts) different angles.
For what length of shot is three angles on each side good enough? You can more or less divide the 3 by the 45 to find the ratio of shot-length reduction to get a number around half a diamond. That means that any 3-angle system -- if strictly and accurately followed -- is only good for hangers.
I think what real people do with "limited-set fractional-ball aiming" systems is to subconsciously fill in the gaps by feeling -- "that looks a little fuller than half ball."