Consider you are playing a game on a billiard table. You have one cue ball that is massless (let's say that can exist in this world).
A massless cue ball will not experience friction and will bounce around and around the table until it lands in a pocket.
Using some principles from physics and geometry, we can investigate how this cue ball aimed at the long side of a billiard table will behave.
As far as physics go, Newton's First law of motion, that an object in motion will stay in motion unless acted upon by another force. Our massless ball is not being acted upon by any forces, except for the force from a cue stick that sends the ball rolling in the first place.
Since our massless ball is not being acted upon by any forces once initially struck by the cue stick, Newton's first law tells us the cue ball will continue along its orginial path in a straight line. Thus, if we imagine the tabletop as a plane, we can represent the cue ball's desired path as a ray on this plane, starting at a point and continuing infinitely in a straight path.
However, the cue ball will eventually hit one of the table walls. What happens then?
Here is where geometry comes into play. Both congruent angles and reflections will be important here.
First, reflections.
The wall of the table will act as a line of reflection on our plane. The ray representing the ball's desired path comes in "contact" with this line of reflection. The portion of the ball's path that lies on the far side of the line of reflection is reflected back into the table. The ball continues on it's path until it comes in contact with another wall of the table where the far portion of the path is reflected back into the table and the pattern continues until the cue ball falls into a pocket.
Next, congruent angles.
Let us call the smallest angle between the ball's path and the line of reflection be called the impact angle. By the vertical angle theorem, the angle between the far portion of the ball's path (before reflection) and the line of reflection is equal to the impact angle, let's call that angle A. Then, since the line of reflection cause - well - a reflection, the angle that the ball bounces at will be congruent to angle A which is congruent to the impact angle. And again, this pattern of reflections and congruent angles will repeat with each bounce of a table wall until the ball comes to rest in a pocket.
Using these simple principles we have discussed, we can aim our cue ball on a plane by creating a grid made up of rectangles that same size as our tabletop. Each intersection on this grid represents a corner pocket (we will ignore side pockets for simplicity in this hypothetical game). Therefore, we can bounce the ball off the table walls and land in a pocket by creating a ray from the ball's initial position to an intersection that lies off our true table top. Each time the ray crosses a vertical or horizontal line of the grid, the ball is actually bouncing off one of the table walls and a reflection is occuring.
You can explore what we learning using this applet. Try moving the ball to different positions on the table and aiming for different pockets by lining the blue aiming point up with an intersection on the grid. The ghost ball will follow the ball's path if the table top extended without walls and the darker ball shows the ball's true path. Notice that the darker ball hits the pocket when the ghost ball reaches the blue aiming point.
Can you experiment and find a way to aim for a side pocket?
References
[1] Urone, P. P., & Hinrichs, R. (2022). 4.2 Newton’s First Law of Motion: Inertia. In College Physics 2e. OpenStax. https://openstax.org/books/college-physics-2e/pages/4-2-newtons-first-law-of-motion-inertia
[2] Pool table geometry. Mathematical Mysteries. (2025, June 9). https://mathematicalmysteries.org/pool-table-geometry/
[3] Perucca, A. (2018, April 24). Arithmetic billiards. Plus Maths. https://plus.maths.org/content/arithmetic-billiards-0
[4] Katok, A. B. (1999, March). Billiard Table as a Playground for a Mathematician. Lecture.
[5] Shelly, I. (2025, July 16). Billiards and math: Understanding the geometry and physics behind prec. Gaming Blaze. https://www.gamingblaze.com/blogs/news/billiards-and-math-understanding-the-geometry-and-physics-behind-precise-cue-shots
[6] Leider, N. (2013). Pool and billiards for dummies. John Wiley & Sons.
[7] Alciatore, D. G. (2017). The illustrated principles of pool and billiards. Sterling.