That was quite a complex document Ekkes o_o
At first glance it seems too technical to be a practical method for learning cut angles,
though I can see you took some time to simplify it and give it a nice presentation.
I thank you for sharing it and for the work you put into it.
But, not trying to be a jerk, does anyone actually learn to cut balls this way?
I've never met one yet.
I think the way we successfully learn cut angles is a lot simpler than
some of the systems and explanations I see online.
At the simplest level, what you are seeing is two circles, a cue ball and an object ball.
From very close it's easy to mentally 'slice' the object ball into fractions,
i.e. 1/3rd ball hit, half ball hit, 1/8th, etc. So when the CB and OB are 3 inches apart,
I can imagine someone figuring out the angle of the shot, then deciding to hit, say, 1/4th of the ball.
Up close, they can then just visually overlap the edge of the cue ball with the 1/4th "mark" on the OB.
But this does not work at longer distances. The object ball is smaller at longer distances.
Visually, a quarter ball hit at 2 feet looks different than a quarter ball hit at 4 feet.
So, you need imagination to figure out a line of aim that will cause the cue ball
*at the moment of impact* to overlap 1/4th of the object ball.
So how do we gain this imagination? Unfortunately I don't think any website can
give it to you. I believe it's simple experience. Our brain remembers familiar shot situations
the same way it remembers familiar faces.
"Oh yeah, that guy with the big black eyebrows is named bob"
"Oh yeah, I remember when I aimed in this direction before, with a very similar cut, the ball went in"
Knowing the overlap (i.e. 1/4 ball at moment of impact), or the number of degrees of the cut,
doesn't actually help your eyes recognize when the big circle correctly overlaps the smaller circle.
Experience tells you that.
The short answer for the OP: practice the cuts you struggle with at a variety of distances,
maybe even on a variety of tables, until they look familiar. Familiar = easier.