Bank Shot Rail Contact Point

The mirror method is the first real system anybody learns for banks: reflect the target through the cushion, aim at the image, and geometry does the rest. It gives $x_c = (x_1y_2 + x_2y_1)/(y_1 + y_2)$, a simple weighted average of the two positions along the rail. It is elegant, it is easy to do in your head, and it is systematically wrong in one direction.

It is wrong because it assumes angle in equals angle out, and the rebound-angle page has just shown that a real cushion sends the ball out WIDER than it comes in. Put $\tan\theta_{out} = \tan\theta_{in}/e$ into the geometry — the incoming path has run $x_c - x_1$ over depth $y_1$, the outgoing has $x_2 - x_c$ over $y_2$ — and one line of algebra gives

\[x_c = \frac{x_1y_2 + e\,x_2y_1}{y_2 + e\,y_1}\]

which is the mirror method with an $e$ attached to every term that belongs to the ball's own side of the rail. Set $e = 1$ and it collapses back to the mirror method exactly, which is the first thing to check about any corrected formula: if it does not reduce to the thing everyone already knows, distrust it.

The size of the correction is what makes this worth having. Take a ball 300 mm off the rail at $x_1 = 500$ mm and a target 600 mm off it at $x_2 = 2000$ mm. The mirror method says aim at 1000 mm. With a real cushion at $e = 0.6$ the answer is 846 mm — a shift of 154 mm, six inches, back toward the shooter. Six inches on a bank shot is the difference between a pot and a rattle, and it is the whole reason mirror-method banks come up short.

Notice which direction the shift goes and why it makes sense: because the ball rebounds wider, it covers the sideways distance to the target more cheaply on the way out, so it needs LESS rail travel on the way in — the contact point moves back toward the cue ball. The flatter the shot, the bigger the correction.

Three honest limits, and none of them is small.

Rail friction is not modelled. On a real table cloth-on-ball friction at the cushion drags the parallel component back, which pushes the answer partway toward the mirror point again. So the true aiming point usually sits between this formula and the mirror one. That is not a reason to ignore the correction — the correction is real and in the right direction — but it is a reason to treat $e$ here as an EFFECTIVE value fitted to your own rails rather than a rubber property. Shoot a few banks you know the geometry of and solve backwards for the $e$ that fits.

English changes everything. Running english makes the rebound longer, reverse english shortens it, and either can move the outgoing path by more than this whole correction. A bank played with sidespin does not obey this equation at all. This is a centre-ball model.

The coordinates are ball centres against the cushion FACE. Not the wooden rail edge, not the diamond line, and not the ball's contact point. A ball frozen to the cushion has its centre one radius out — 28.6 mm on a pool table — so a shot planned from the rail rather than from the ball's centre starts more than an inch wrong. The diamond systems that professionals use are calibrated empirically to absorb this and everything else, which is why they are stated as diamond counts rather than as geometry: they are a fitted model of a real table, and this is the physics that fitted model is approximating.

Use this as a way of understanding WHY the corrections your eye already applies are the size they are, rather than as a number to compute at the table. Nobody is going to solve a rational function between shots. But knowing that the correction always goes the same way, and grows as the shot flattens, is worth carrying around — and having the equation lets you test your own cushions honestly instead of guessing at them.