Many pool playing systems, namely aiming systems, use different segments of a ball for alignments such as edges, centers and quarters. The edges are pretty easy to obtain, as is the center. The quarters however, are not quite as obvious. Most systems assume that “eye-balling” the quarter is going to be precise enough, and with practice you can learn to “just see it”. But, is there a mathematical way to know the true quarter? Yes there is.
Lets look at the right quarter. When you look at the face of a ball, the 12:00 and 3:00 lines are very easy to visualize: straight up and straight right. The line that is exactly between those is 1:30. Again, very easy to visualize. Now if you imagine a line connecting 12:00 and 3:00 (forming a triangle), it bisects the quarter ball segment. Likewise, this line bisects the 1/8th ball segment, as you can see below.
Although you can use the exact bisecting lines to find the quarters, I like to just look at the perimeter of the ball. If you look at the red dot at the 1:30 mark, you can see how far up the quarter mark red dot is, and also how far down the 1/8th mark red dot is. I memorize these distances, which only takes a handful of shots to get the hang of. Either method works, it is personal preference which one you like.
It is also useful to know that on a modern stripe ball, the stripe edge is 1/16th of an inch outside the true quarter. Old centennial stripe balls are exactly on the quarter.