Master the art of the bank shot with this calculator. Based on the principle of angle of incidence equals angle of reflection, this tool helps you determine the optimal rail contact point to sink your ball into the target pocket.
Top Rail (Y=100)
Bottom Rail (Y=0)
Left Rail (X=0)
Right Rail (X=50)
Calculation Results:
function calculateBankShot() {
var cueBallX = parseFloat(document.getElementById("cueBallX").value);
var cueBallY = parseFloat(document.getElementById("cueBallY").value);
var targetPocketX = parseFloat(document.getElementById("targetPocketX").value);
var targetPocketY = parseFloat(document.getElementById("targetPocketY").value);
var railChoice = document.getElementById("railChoice").value;
var tableWidth = 50; // Standard 9-foot table playing surface width in inches
var tableLength = 100; // Standard 9-foot table playing surface length in inches
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = ""; // Clear previous results
// Input validation
if (isNaN(cueBallX) || isNaN(cueBallY) || isNaN(targetPocketX) || isNaN(targetPocketY)) {
resultDiv.innerHTML = "Please enter valid numbers for all position inputs.";
return;
}
if (cueBallX tableWidth || targetPocketX tableWidth) {
resultDiv.innerHTML = "All X-positions must be between 0 and " + tableWidth + " inches.";
return;
}
if (cueBallY tableLength || targetPocketY tableLength) {
resultDiv.innerHTML = "All Y-positions must be between 0 and " + tableLength + " inches.";
return;
}
var railContactX, railContactY;
var reflectedPocketX, reflectedPocketY;
var errorMessage = "";
switch (railChoice) {
case "top":
// Reflect target pocket across the top rail (Y = tableLength)
reflectedPocketX = targetPocketX;
reflectedPocketY = tableLength + (tableLength – targetPocketY);
// Check if target pocket is on the chosen rail, which is not a bank shot off that rail
if (targetPocketY === tableLength) {
errorMessage = "The target pocket is on the top rail. This is a direct shot, not a bank shot off the top rail.";
break;
}
// Check if object ball is on the chosen rail
if (cueBallY === tableLength) {
errorMessage = "The object ball is on the top rail. This is not a bank shot off the top rail.";
break;
}
// Calculate intersection point on the top rail (Y = tableLength)
// Line equation: (y – cueBallY) / (reflectedPocketY – cueBallY) = (x – cueBallX) / (reflectedPocketX – cueBallX)
// Set y = tableLength
if (reflectedPocketY – cueBallY === 0) { // Avoid division by zero if line is horizontal
errorMessage = "Cannot calculate: Object ball and reflected pocket are on the same Y-level. This implies a straight shot parallel to the rail or an impossible reflection.";
break;
}
railContactX = cueBallX + (reflectedPocketX – cueBallX) * (tableLength – cueBallY) / (reflectedPocketY – cueBallY);
railContactY = tableLength;
if (railContactX tableWidth) {
errorMessage = "The calculated rail contact point (" + railContactX.toFixed(2) + " inches) is off the top rail. This shot is not feasible as a single-rail bank shot to this pocket.";
}
break;
case "bottom":
// Reflect target pocket across the bottom rail (Y = 0)
reflectedPocketX = targetPocketX;
reflectedPocketY = 0 – targetPocketY;
if (targetPocketY === 0) {
errorMessage = "The target pocket is on the bottom rail. This is a direct shot, not a bank shot off the bottom rail.";
break;
}
if (cueBallY === 0) {
errorMessage = "The object ball is on the bottom rail. This is not a bank shot off the bottom rail.";
break;
}
// Calculate intersection point on the bottom rail (Y = 0)
if (reflectedPocketY – cueBallY === 0) {
errorMessage = "Cannot calculate: Object ball and reflected pocket are on the same Y-level. This implies a straight shot parallel to the rail or an impossible reflection.";
break;
}
railContactX = cueBallX + (reflectedPocketX – cueBallX) * (0 – cueBallY) / (reflectedPocketY – cueBallY);
railContactY = 0;
if (railContactX tableWidth) {
errorMessage = "The calculated rail contact point (" + railContactX.toFixed(2) + " inches) is off the bottom rail. This shot is not feasible as a single-rail bank shot to this pocket.";
}
break;
case "left":
// Reflect target pocket across the left rail (X = 0)
reflectedPocketX = 0 – targetPocketX;
reflectedPocketY = targetPocketY;
if (targetPocketX === 0) {
errorMessage = "The target pocket is on the left rail. This is a direct shot, not a bank shot off the left rail.";
break;
}
if (cueBallX === 0) {
errorMessage = "The object ball is on the left rail. This is not a bank shot off the left rail.";
break;
}
// Calculate intersection point on the left rail (X = 0)
if (reflectedPocketX – cueBallX === 0) { // Avoid division by zero if line is vertical
errorMessage = "Cannot calculate: Object ball and reflected pocket are on the same X-level. This implies a straight shot parallel to the rail or an impossible reflection.";
break;
}
railContactY = cueBallY + (reflectedPocketY – cueBallY) * (0 – cueBallX) / (reflectedPocketX – cueBallX);
railContactX = 0;
if (railContactY tableLength) {
errorMessage = "The calculated rail contact point (" + railContactY.toFixed(2) + " inches) is off the left rail. This shot is not feasible as a single-rail bank shot to this pocket.";
}
break;
case "right":
// Reflect target pocket across the right rail (X = tableWidth)
reflectedPocketX = tableWidth + (tableWidth – targetPocketX);
reflectedPocketY = targetPocketY;
if (targetPocketX === tableWidth) {
errorMessage = "The target pocket is on the right rail. This is a direct shot, not a bank shot off the right rail.";
break;
}
if (cueBallX === tableWidth) {
errorMessage = "The object ball is on the right rail. This is not a bank shot off the right rail.";
break;
}
// Calculate intersection point on the right rail (X = tableWidth)
if (reflectedPocketX – cueBallX === 0) {
errorMessage = "Cannot calculate: Object ball and reflected pocket are on the same X-level. This implies a straight shot parallel to the rail or an impossible reflection.";
break;
}
railContactY = cueBallY + (reflectedPocketY – cueBallY) * (tableWidth – cueBallX) / (reflectedPocketX – cueBallX);
railContactX = tableWidth;
if (railContactY tableLength) {
errorMessage = "The calculated rail contact point (" + railContactY.toFixed(2) + " inches) is off the right rail. This shot is not feasible as a single-rail bank shot to this pocket.";
}
break;
default:
errorMessage = "Invalid rail choice.";
}
if (errorMessage) {
resultDiv.innerHTML = "" + errorMessage + "";
} else {
resultDiv.innerHTML = "To execute this bank shot, aim to hit the " + railChoice + " rail at approximately:" +
"X-Position: " + railContactX.toFixed(2) + " inches" +
"Y-Position: " + railContactY.toFixed(2) + " inches" +
"(Note: X=0 is left rail, X=" + tableWidth + " is right rail. Y=0 is bottom rail, Y=" + tableLength + " is top rail.)";
}
}
Understanding the Billiards Bank Shot
A bank shot in billiards (or pool) is a shot where the object ball is hit into one or more rails before it goes into a pocket. It's a fundamental skill that allows players to pocket balls that are not directly accessible or to achieve better cue ball position for the next shot. The success of a bank shot relies heavily on understanding the physics of reflection.
The Physics of Reflection: Angle of Incidence
The core principle behind a bank shot is that the angle at which a ball strikes a rail (the angle of incidence) is equal to the angle at which it leaves the rail (the angle of reflection). Imagine a line perpendicular to the rail at the point of contact; the angle between the incoming path of the ball and this perpendicular line is the angle of incidence, and the angle between the outgoing path and this perpendicular line is the angle of reflection. These two angles are always equal, assuming a perfect elastic collision and no spin on the ball.
How This Calculator Works
This calculator uses a simplified 2D coordinate system for a standard 9-foot pool table playing surface (100 inches long x 50 inches wide). The origin (0,0) is at the bottom-left corner of the playing surface. The X-axis runs along the bottom rail (width), and the Y-axis runs along the left rail (length).
To find the optimal rail contact point, the calculator employs a geometric trick: it reflects the target pocket's position across the chosen rail. Then, it calculates a straight line from the object ball's position to this 'reflected' pocket. The point where this straight line intersects the chosen rail is the ideal contact point for the bank shot, satisfying the angle of incidence equals angle of reflection principle.
Input Definitions:
Object Ball X-Position (inches): The horizontal coordinate of the object ball on the table. (0-50 inches)
Object Ball Y-Position (inches): The vertical coordinate of the object ball on the table. (0-100 inches)
Target Pocket X-Position (inches): The horizontal coordinate of the center of the target pocket. (0-50 inches)
Target Pocket Y-Position (inches): The vertical coordinate of the center of the target pocket. (0-100 inches)
Rail to Hit: Select which rail you intend for the object ball to strike first.
Output Explained:
The calculator provides the X and Y coordinates on the chosen rail where the object ball should ideally make contact. These coordinates represent the precise spot on the rail that will redirect the ball into the target pocket, assuming a perfect shot with no spin.
Important Considerations and Limitations:
Spin (English): This calculator assumes a perfectly struck ball with no spin. In reality, applying spin (English) to the cue ball or object ball will alter the angle of reflection.
Friction and Rail Elasticity: Real-world rails are not perfectly elastic, and friction exists, which can slightly affect the reflection angle.
Object Ball vs. Cue Ball: This calculator determines the path for the *object ball* to hit the rail and then the pocket. The inputs for "Object Ball Position" refer to the object ball's starting point. You must then hit the object ball such that it travels to the calculated rail contact point.
Table Conditions: Factors like table cloth condition, rail wear, and temperature can subtly influence shot dynamics.
Single Rail Shots: This calculator is designed for single-rail bank shots. More complex shots involving multiple rails require advanced calculations.
Pocket Size: The calculator assumes a point target for the pocket. In reality, pockets have an opening, offering a margin of error.
Example Scenario:
Let's say your object ball is at X=20 inches, Y=30 inches. You want to bank it off the top rail into the bottom-middle pocket (X=25 inches, Y=0 inches).
Set "Object Ball X-Position" to 20.
Set "Object Ball Y-Position" to 30.
Set "Target Pocket X-Position" to 25.
Set "Target Pocket Y-Position" to 0.
Select "Top Rail (Y=100)" from the "Rail to Hit" dropdown.
Click "Calculate Bank Shot".
For these inputs, the calculator will determine the optimal contact point on the top rail. The result would be approximately X=22.06 inches on the top rail (Y=100).