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© Copyright 1999, Jim Loy
This article is now in three parts: Part I (calculating), Part II (complications), and Part III (putting it all together).
In pool, a bank shot is when you drive an object ball into a rail, and then into a pocket, somewhere on the other side (or end) of the table. A kick shot is when you drive the cue ball into the rail, and then into the object ball. The diagram shows a typical kick. Pool teachers will tell you that the angle that the cue ball goes into the rail is equal to the angle that it leaves the rail (at the point labelled Y, in the diagram), just like a light ray reflecting off a mirror. For simplicity's sake, this is true. But, there are quite a few complications, with which we will deal later (see Part II (complications)). Most of these diagrams show kicks. But most of this applies just as well to banks.
Why do I show so many methods? Well, I find this all interesting and informative. You can try a few and pick the methods that you like. No one method is perfect. Some are easier to use in different situations. Also, once you have chosen a point of aim, you may want to check it, perhaps with method #4 (trial and error).
Warning: During practice, you may want to place your chalk on the aiming point, or even make chalk marks on the table. Do not do this in tournaments, as that may be a foul. In some tournaments even setting your cue down on the table, for the purpose of measuring, is a foul.
Method #1 (calculate): Where do you aim, to get these equal angles? If the cue ball and the target ball are the same distance from the rail, then you just aim half-way between them. What about the situation in the diagram? Well, similar triangles lead us to this formula: y=ax/(a+b). So, will they let you use a ruler and a calculator in a pool game? Well, first of all, the ruler is right there on the rail. There are little pearly diamonds (dots, more often) on the rail, to help you with this situation. You don't measure inches, you measure diamonds (and fractions of a diamond). And there are other ways to determine where point Y is, as you will see below.
Method #2 (X method): This method is in Willie Hoppe's book. You mentally draw the X in this diagram, from the cue ball to the "image" of the target ball on the rail, and from the target ball to the "image" of the cue ball on the rail. These two lines intersect somewhere between the two balls (and toward the rail from the two balls). Your target point Y is the image of this intersection point. Unless the two balls are a long distance apart, this method works very simply. Incidentally, geometry shows that the point Y, determined in this way, is the same exact point Y, determined above using similar triangles (or using our formula).
Method #3 (parallel):Yet another method, this one advocated by Dr. Cue, is to find the midpoint between the two balls. You can imagine drawing a line between the two balls, and finding its midpoint. Then aim along the line between this midpoint and the image of the object ball. The actual cue ball path (from the cue ball to point Y) will be parallel to this line. With a little practice, you can easily move your cue parallel to this line. This method works very well. It tends to be very inaccurate, if the balls are a long distance apart (because judging parallel lines is error prone). Incidentally, geometry shows that the cue ball path, determined in this way, goes to the same point Y, determined above using other methods.
Method #4 (trial & error): A fourth method (trial and error) works fairly well, when the balls are a long distance apart. Estimate where point Y might be. Then find a line between this estimate of Y and the object ball. Find the point on this line that is the same distance from the rail as the cue ball (distance a from the rail, in the first diagram). Find the image of this point on the rail (Z in the diagram). If you guessed right about where Y should be, then Y is exactly halfway between points X and Z, in the diagram. If it is not, then the midpoint between X and Z is your new estimate of where Y is. Repeat the above process until your Y is really close to the right place. If you practice this, you may figure out a shortcut, by adjusting where you put new estimates of Y, partway through this process. It works fairly easily and smoothly, once you get used to this method. This method is particularly easy, if the cue ball rests on the opposite rail. Then you do not have to guess about your distance (a) from Z. This method can be used to fine tune some of the other methods which do not work so well when the balls are far apart. Sometimes this method may not seem very much like "trial and error", as you may zero in on the correct aiming point right away.
Here is a diagram that illustrates the above method. I have intentionally chosen a bad estimate of Y, and have called it Y1. By imagining a line between Y1 and the target (ball or pocket), I can find the point Z1. The next estimate of Y (Y2) is half way between Z1 and X. This is a much better guess for Y. I can repeat the above steps: find a new Z (Z2). The new Y (Y3) is half way between X and Z2. And so on. You can see why I call it "trial and error."
Method #5 (designator tracks): A couple of people have called these "designator tracks." I do not know just how widespread this terminology is. Essentially, you already know some of the tracks on the pool table. For example, you know the two banks shown in this diagram, from the second diamond and from the fourth diamond, to the appropriate diamonds on the far side of the table. We want to bank our object ball in a similar manner, but it does not lie on one of the designator tracks. So we aim it roughly parallel to the nearest designator track. We know that these lines are not really parallel. So we adjust our aim slightly. Designator tracks work for any number of rails, even 6 and 7 rail kicks. You just need to know a track that works, in order to shoot a similar shot from a nearby place on the table.
Method #6 (spot on the wall):This method can be used with any of the above methods, or by itself. Instead of aiming at a diamond, you aim at a spot somewhere distant from the table. This spot is roughly the width of a table beyond the spots. Somehow you have determined where this spot is, for one of the pockets. Maybe you did this by trial and error. Maybe you sighted along the designator tracks of the previous method (as I show in this diagram). Then all banks off the same rail into the same pocket should be aimed at the same spot. This spot is an image of the target pocket, as if you had a mirror set up over the rail. This method is rather amazing, as it can be used to take into account all of the complications of Part II of this article. In other words, you can adjust the position of the spot on the wall to take into account ball speed, english, draw, almost anything. It can also be used for two and three-rail kicks and banks. It is usually used to aim at a target pocket, not a target ball. But, there are two or three ways to use it to aim at a target ball, as well. One of these ways is the next method, the ghost ball. By the way, make the spot something that doesn't move. If you aim at that girl's nose, you will miss when she moves.
Method #7 (ghost ball): This one works well if the angle is shallow (nearly parallel to the rail). Imagine a ghost ball off the table, about the same distance outside the rail that the target ball is inside the rail. If the angle is shallow, you may even be able to adjust your hit on the object ball in order to make it go where you want (like into a pocket), accurately. With this method, you can deal with the complications (which I cover below) in two different ways. You can deal with them in the same way that you do with the other methods. Or you can make your ghost ball farther outside the table, through point Y' (which I also describe later). Some adjustment has to be made, depending on how far the object ball is from the rail. Experiment, and let me know if you hit upon a good method. You can actually measure the position of the ghost ball, with your cue or fingers.
Method #8 (trial & error backwards): This is the backward version of Method #4, above. And it should work about as well, in most situations. Guess where the point Y is going to be. Find a point the on your cueball's track that is the same distance from the rail as the target ball (or pocket). Then find point Z on the rail, and find a new Y midway between Z and X. Repeat if you were not very close the first time. Other methods (above) can be done backwards. But they are normally not very handy that way. This one is not bad, though.
Method #9 (judgment): I should have mentioned this one first. It works well on many banks and kicks. Especially, it works when the target ball is near the rail that you are kicking to. With practice, you can get a nearly perfect hit with judgment, when calculation will not help. The shot in the diagram is worth practicing. You should actually make the ball about half the time, in my opinion. It is tougher if you need side english to get by the object ball. And beware of the scratch, if you have to cut the object ball after the kick. It takes practice to develop the needed judgment. So, practice this shot. And take some time lining it up.
Method #10 (fractions):Here is an unusual method that I like. If we are banking cross corner, there is a fraction associated with each of the various imaginary lines that are parallel to the side rails. This fraction determines how far down table that you aim. In the diagram, the red ball is approximately on the 1/3 line (midway between the side rails), and so the aiming point is 1/3 of the way toward the end rail (y=x/3). The diagram shows the fractions associated with the three different diamonds on the end rail. The bottom side rail (in the diagram) fraction is 1/2, as you would expect. Those four fractions will get you to within a half diamond of almost any place on the table. You can adjust parallel from there, for the in-between points. This method is a simple application of method #1 (calculate) above. In case you are curious, the half-diamond fractions (starting at the bottom side rail in the diagram) are 1/2, 7/15, 3/7, 5/13, 1/3, 3/11, 1/5, 1/9. The unit fractions (1/3, 1/5, and 1/9) are easier to use than the others. But, none of this is actually difficult. At least try the 1/3 shots, and maybe make it one of the "designator tracks" (see method #5 above) that you know.
By the way, other kicks on the table have other fractions associated with them. Some are simple (like 2/3). But some are not. Few are more complicated than the 3/7 line above. But, it is probably not worth it to memorize very many of them.
Method #11 ("parallel points"): In Desmond Allen's One Rail, Two Rails, Three Rails and More! (and its clone Win at Pocket Billiards, which I bought thinking it was a different book), we find this intriguing method, which took me a while to decipher. We are going to bank the red ball into the corner as shown. Dr. Allen says, "Find a spot on the table near the desired point of contact that is parallel to the (red) ball." Wow, that sentence makes little sense to me. He was talking about kicking the cue ball, so where I wrote "(red) ball" he wrote "cue ball." I wrote to Dr. Allen about that sentence, and received no response. Luckily his diagram is fairly informative, and, I have some idea what pool players mean when they say that a point is parallel to a ball (if you are a mathematician, you probably just pulled out your last remaining hair). In my diagram, we want to find a point A, over which we think the red ball may roll after it banks, the same distance from the side rail as the red ball. I think that is what he means in his sentence. Then find the point B halfway between the Red ball and the point A. The image of point B, on the rail, is C. Now we have a (red) line AC. We lay our cue stick (black line in the diagram) on the table through the target pocket (the lower left corner in the diagram), and parallel to the line AC. This gives us a point D on the rail, as in the diagram. Our point of aim is midway between D and C. Dr. Allen says that this method is very accurate. Actually, its accuracy depends on how well you choose your point A. In the diagram, it is helpful if the red line is very close to the black line. If the ball you are banking is near the rail (top rail in this example) and A is chosen poorly, this method can be way off. But normally, it is easy to make a good guess about where to put point A. So it is accurate. Also, you may end up with point A on the other side of the black line, which is no problem.
Method #12 (rail track): Jack Koehler, in The Science of Pocket Billiards gives this as the "rail track system." It is easier than many of the above methods. Here we are banking the red ball cross corner. We imagine a point halfway between the red ball and the rail behind the ball (at the bottom of the diagram). The book says that the line is perpendicular to the ball; actually it is perpendicular (at right angles) to the rail. Then we imagine a line from this point and the pocket in the upper left of the diagram (probably using the cue as this line) Where that line meets the rail behind us (bottom of the diagram) is the back end of our ball path to the side rail (top of the diagram). Geometry shows that (if done in a similar way) this produces the same aiming point as method #1 above. This method is less accurate the closer the ball is to the rail that it will hit (top rail in the diagram), and more accurate the farther it is from this rail. Most of the above methods are more accurate the closer the ball is to that rail. So this method may be preferable to those methods, when far from that rail.
Method #13 (eyeball): An important method is to just eyeball the two angles off the rail, and estimate an aiming point that will give you equal angles. This is essentially the same as method #9 (judgment), except that we are concentrating on the angles, and not on other factors like the path of the balls. In my opinion, this is hard to do.
Method #14 (backwards): With any of the above methods, instead of calculating how to kick the cue ball into the red ball (or bank the red ball into a pocket) you can pretend that you are kicking the red ball into the cue ball (or kicking the pocket into the red ball), and find the same aiming point. Most of the time, this is no easier than calculating forwards. But occasionally, it may be easier, or produce a more accurate result. An example is when the cue ball is a great distance from the rail and the object ball is closer to the rail, some of the above methods produce a relatively accurate way to find the aiming point from the object ball to the rail to the cue ball. The cue ball would be a small target, so this idea seems inaccurate. But the aiming point for hitting the object ball can be determined very accurately, in this way.
Diamonds: In real life, we do not aim at the rail contact point Y. And, in Part II (complications), we will find out why. Instead of aiming at Y, we aim at a point short of Y, a point on the wooden part of the rail, and not the rubber part of the rail. In my diagrams, this actual aiming point is labeled Y'. It is either one of the diamonds (those pearly diamonds or dots on the wooden part of the rail) or at an imaginary diamond between the actual diamonds.
In every one of the methods that I described above, you can find this aiming point (Y'), because it is the image of the calculated point Y, inset at the same distance as the diamonds.
Go to Part II (complications), which covers deadness of the rail, the fact that balls are not points, side english, draw & follow, speed, rail compression, and shooting near a rail.
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