A geometric analysis of the Diamond System

Publié par Jean-Pol.G le 1 Mars 2025

3-Cushion Bank Shot, Diamond System

(Click on the graphics to enlarge)

Introduction

This article is a geometric analysis of the "Diamond System".

Thanks to its simplicity, and despite its imperfections, this system is widely used. Despite efforts, I could not find any trace of its origins, its creators, or justification for the method. I have searched the Internet, questioned top players like Raymond Ceulemans and Francis Connesson, a post on AzBilliards.com rolled up interest but did not provide the expected feedback.

In this article, I will try to explain from where the principle of the arithmetic formula is coming and how the rails codes may have been conceived.

This is targetting readers familiar with the system who have once day wondered from where the magic formula was coming. We will not recall it, nor will we point out its imperfections nor suggest adjustments. We will merely attempt to find a geometric justification for it, which, according to me, derives from the principle of "2-Cushion Bank Shot", which we will now examine in detail.

Preliminaries

The explanations are based on geometric properties in a different context from the game itself.

In the mathematical model, the balls are points, with no diameter. The principle "Reflection = Incidence" is applied, taking into account the side spin according to the rule: "one tip of spin = one diamond of deviation in width", "one tip of spin = two diamonds of deviation in length". We assume, which is unrealistic, that the spin applied at the shot remains constant throughout the entire trajectory. We also assume that contact with the rails does not cause any hooping, and does not communicate any additional spin.

Warning : Of course, the reality of the game is different and the model is only an approximation of it. Nevertheless we will try to get geometric explanations for the formula and for the codes of the Diamond System. In particular, we will explain why the markers on the starting cushion go from 5 to 5 while that of the aiming cushion go from 10 to 10.

In the following, the (geometric) points of diagrams and examples are identified by their coordinates in the Cartesian system whose axes are the bottom long rail (x's) and the short left rail (y's):

The origin (0,0) is at the bottom left corner of the table. The coordinates of the top left corner are (0,4), those of the bottom right corner (8,0) and those of the last corner (8,4).

The spin is taken into account according to:

I- 2-Cushion Bank Shot

The "2-Cushion Bank Shot" recipe is as follows:

Playing with no side spin, if we hit at a distance "d" from the corner, we end on the third rail at a distance "2d" from the starting point.

We will justify this rule when the shot makes a 45-degrees angle with the rails and when the starting point and the aim are on two parallel rails. We will also see that outside these two cases, the rule is inaccurate.

Consider in the adjacent figure the triangles

ACA', ACR

They are equal and right-angled at C. All green angles are equal due to the "Reflection = Incidence" principle. The lines SA and FR are parallel. The distance SF is equal to the distance A'R. It follows that

The distance SF is twice the distance CR

Note that to be able to state that SF is twice CR, it is necessary that SF and CR are parallel.
Note as well that SF is twice CA only in case the triangle ACR is isosceles, that is, if the direction SA is 45-degrees with respect to the cushions.
In the figure above, imagining the starting point is F and end is S. R and S being on two parallel cushions, the distance SF being 4 diamonds, we aim R, 2 diamonds from C.
In the figure above, imagining the starting point is S and end is F, 4 diamonds from S. We must not aim the short cushion 2 diamonds from the corner because the aim and the starting point are not on two parallel cushions and because SA is not 45-degrees.
To determine the aim (no spin), we note that the median MC of the triangle SCF is parallel to the lines SA and FR. In game conditions, this property gives a way to determine the aim A, because it is quite easy to visualize the median MC and its parallel SA.

Illustration

Example 1- Start long cushion is S = (3,0), expected long cushion finish is F = (5,0), distance(S,F) = 2. The recipe says we should aim short cushion at distance(A,C) = 1, so A = (0,3). SA is 45-degrees. The result is correct.
Example 2- Start long cushion is S = (4,0), expected long cushion finish is F = (6,0), distance(S,F) = 2. Incorrect use of the recipe says we should aim short cushion A at distance(A,C) = 1, so A = (0,3). SA is not 45-degrees, SF is not parallel to CA. The result is incorrect.
Example 3- : In conditions of example 2- choose aim short cushion A at distance(A,C) = 1 and add spin. The result is correct.
Example 4- In conditions of example 2-. Choose aim short cushion A with median, so A = (0,3.2). No spin. The result is correct.
Example 5- Start long cushion is S = (6,0), expected long cushion finish is F = (4,0), distance(S,F) = 2. The recipe could be used because SF and CA are parallel. Choose long rail aim A = (1,4), distance(A,C) = 1. The result is correct.
Example 6- Start short cushion is S = (8,1), expected short rail finish is F = (8,3), distance(S,F) = 2. With recipe choose aim A = (0,3). SF and CA are parallel. The result is correct.

II- 3-Cushion Bank Shot

Example 7- Suppose we want to play no spin 3-Cushion Bank Shot the opposite diagram . The start is S = (6,0). By guesswork, we see that we could end on the 3rd rail two diamonds from the starting point at F' = (4,0). With the "2-Cushion Bank Shot" recipe, S and A being on parallel cushions, aim exactly A = (1,4). The result is good.
Example 8- Suppose we want to play no spin 3-Cushion Bank Shot the diagram opposite. The start is S = (4,0). By guesswork, we see that we could finish on the 3rd cushion two diamonds in front of the starting point at F' = (6,0). Using the direction of the "median", we aim A = (0,3.2). Finish is F' = (4,0) as desired.

III- Diamond System

We assume that the reader is familiar with the (corner 50) "Diamond System". (Possibly look at the annexes). We expect to understand its codes and its calculation formula.

Example 9- Let's start at corner S = (8,0) = Diamond 50, and aim long cushion A = (3,4) = Diamond 30. Let's play without spin.

We hit the long rail (3rd rail) F' = (2,0) = Diamond 20, 6 diamonds from the start, according to the "2-Cushion Bank Shot" recipe, S and A being on parallel rails.

We can note that the lack of spin makes us end on the 4th rail at F = (7,4) = Diamond 10.

Conversely, starting S = (8,0) = Diamond 50, let's assume that our objective is finish 3rd rail F' = (2,0). We see that for that, according the "2-Cushion Bank Shot" recipe, we have to aim A = (3,0).

We note the relation distance(C,A) = distance(S,F')/2.

With unit "1 diamond = 5" on the starting rail and "1 diamond = 10" on the aiming rail, with code 10 for the corner (0,0) on the start rail, and code 0 for the corner (0,4) on the aiming rail, the formula becomes

A(30) = S(50) - F'(20).

and the codes

Start	10	15	20	25	30	35	40	45	50
Aim	0	10	20	30	40	50	xx	xx	xx

That is the Diamond codes for Start and Aim. With these, F', being the arrival on the 3rd rail, the aim is given by A = S - F'

We got a formula, codes for the start rail and codes for the aim rail. We got a step towards the Diamond system. But we have not yet achieved anything on the essential, namely the end on 4th rail.

A first system that does not work

Let's play no spin.

Start S = (8.0) = Diamond 50, aim A = (2.65.4) = Diamond 26.5. The 3rd rail finish is F' = (2.35.0) and the 4th rail finish is F = (8.4) = Diamond 20.

We could decide to code this corner F by 23.5. Therefore the formula A = S - F would give the correct aim and the desired result for the start S = (8,0) = Diamond 50.

Unfortunately, the use of the formula and this code, suitable for the previous start S = 50, is no longer suitable for start at S = (7,0) = Diamond 45, nor for S = (6,0) = Diamond 40, nor for S = (5,0) = Diamond 35 .... The 4th cushion finish F moves further and further away from the corner. The 3rd rail finish F' does not vary, but the finish lines F'F turn noticeably.

Now let's play full side spin.

Example 10- Start S = (8,0) = Diamond 50, aim A = (3,4) = Diamond 30, spin = 2.7.

The 3rd rail end is a little right of (2,0) = Diamond 20. The spin produces its effect and 4th cushion end is F = (8,4) = Diamond 20.

We code the corner F = (8,4) = Diamond 20 by 20. The formula A = S - F gives the correct aim.

We can highlight two points

By varying starting points, aiming according to A = S - F(20), the finish lines F'F are very stable. And that it is the same for arrivals other than F = Diamond 20.
The spin does not affect much the arrival F' on the 3rd cushion because there is compensation between elongation on the aim rail and shortening on the 2nd rail.