Billiards - Free online pool and billiards game

This is an open-source project bringing unsophisticated billiards physics written in TypeScript to the browser.

Demos run in all major desktop and mobile browsers and use WebGL

• Nine ball ⬀ make a break and share replay link with friends
• Three cushion billiards ⬀ the ultimate test of physics and player (average on both counts). You need the beginner mode ⬀. Try the practice mode ⬀ to drill specific shots.
• Sagu ⬀ Korean four-ball carom game. Try sagu on a 5ft table ⬀.
• Snooker ⬀ we await the first 147 submission to the leaderboard.
• 8-Ball ⬀ try it in Hi-Res mode!
• Play vs the Claw ⬀ and TheFarJaw ⬀.
• Try Multiplayer online pool lobby.
• Try to get on the leaderboard of highest breaks hosted on Vercel.
• Inspect physics and tweak constants using diagrams.
• Set up trick shots and practice ⬀.
• Tune three cushion physics ⬀ and optimise ⬀ parameters
• Parallel parameter ⬀ tuning using Web Workers ⬀.
• Three cushion trainer and sensitivity analysis ⬀ from an excellent fork ⬀ of this project.
• Another great fork for Italian 5 pin.
• Generate SVG and PNG diagrams of shots for blog posts.
• Race again the clock to clear the table in a speedrun.
• Customise your cue and emblem for other players to see
• Sit a hand curated exam to asses your level of play.
• Invite friends and play to win as many games as you can in 30 mins in hourly arenas (lichess.org style)

• Backspin, sidespin and cushion bounces well modeled.
• Presentation using WebGL in any modern browser on mobile, linux, mac or windows.
• Record and playback breaks.
• Two player online mode with nchan nginx server.
• Nine ball, snooker, three cushion and Sagu billiards rules.
• Deploys to GitHub Pages, Vercel and Render with GitHub Actions.
• Runs on and was developed mostly on a potato e.g. Raspberry Pi 4.

You dont, you just play in your browser at billiards.tailuge.workers.dev/lobby

If you prefer, you can install a thin client wrapper:

• Windows / macOS / Linux — Download
• Android — Driect Download or via APKPure or Obtainium

• Papers on ball mechanics Han 2005 with important corrections by Kiefl.
• cushions, max spin, simulation and constants 1 2 3 4 5
• 3D graphics uses three.js
• Inline LaTeX editor for equations in README.md

Based on Han 2005 paper

$$\vec{v}_a = \vec{v} + (\vec{up} \times R\vec{\omega})$$

$$\dot{v} = -\mu g \frac{\vec{v}_a}{|\vec{v}_a|}$$

$$\dot{\omega} = -\frac{5}{2}\frac{\mu g}{R} \frac{\vec{v}_a}{|\vec{v}_a|}$$

$$\dot{\omega}_z = -\frac{5}{2}\frac{M_z}{mR^2} \text{sgn}(\omega_z)$$

$$\dot{v} = -\frac{5}{7}\frac{M_{xy}}{mR} \frac{\vec{up} \times \vec{\omega}}{|\vec{\omega}|}$$

$$\dot{\omega} = -\frac{5}{7}\frac{M_{xy}}{mR^2} \frac{\vec{\omega}}{|\vec{\omega}|}$$

where

$M_{xy} = \frac{7}{5\sqrt{2}} R \mu m g$ , $M_z = \frac{2}{3} \mu m g \rho$

Based on paper by Alciatore incorporating throw effect due to the small amount of friction between balls. Figures to prove consistency between the code and paper can be found in the Mathavan model validation diagram.

For ball $a$:

$$\vec{v}_a \leftarrow \vec{v}_a + \frac{J_{\text{normal}}}{m}\hat{n} + \frac{J_{\text{tangential}}}{m}\hat{t}$$

$$\vec{\omega}_a \leftarrow \vec{\omega}_a + \frac{1}{I} (\vec{r}_a \times \vec{J}_{\text{tangential}})$$

For ball $b$:

$$\vec{v}_b \leftarrow \vec{v}_b - \frac{J_{\text{normal}}}{m}\hat{n} - \frac{J_{\text{tangential}}}{m}\hat{t}$$

$$\vec{\omega}_b \leftarrow \vec{\omega}_b + \frac{1}{I} (\vec{r}_b \times \vec{J}_{\text{tangential}})$$

Where:

The relative velocity at the point of contact is computed as:

$$\vec{v}_{\text{rel}} = (\vec{v}_a - \vec{v}_b) + \vec{r}_a \times \vec{\omega}_a - \vec{r}_b \times \vec{\omega}_b$$

$$\vec{v}_{\text{slip}} = \vec{v}_{\text{rel}} - (\vec{v}_{\text{rel}} \cdot \hat{n}) \hat{n}$$

$\vec{r}_a = -R \cdot \hat{n}$ and $\vec{r}_b = R \cdot \hat{n}$

$J_{\text{normal}} = \frac{-(1 + e)v_{\text{rel,normal}}}{(2/m)}$

$J_{\text{tangential}} = \min\left( \frac{\mu J_{\text{normal}}}{v_{\text{rel}}}, \frac{1}{7} \right)(-v_{\text{rel,tangential}})$

$\hat{n}$: normal unit vector along the line of centers.

$\hat{t}$: tangential unit vector perpendicular to $\hat{n}$.

This is based on a paper by Mathavan. Many of the figures from the paper are recreated to confirm correctness.

Slip velocity at cushion contact point I

$$ ẋ_I = \dot{v_x} + \dot{\omega_y} R \sin \theta - \dot{\omega_z} R \cos \theta \qquad ẏ'_I = -\dot{v_y} \sin \theta + \dot{\omega_x} R $$

$$ \phi = \arctan\left(\frac{ẏ'_I}{ẋ_I}\right) \qquad s = \sqrt{(ẋ_I)^2 + (ẏ'_I)^2} $$

Slip velocity at table contact point C

$$ ẋ_C = \dot{v_x} - \dot{\omega_y} R \qquad ẏ_C = \dot{v_y} + \dot{\omega_x} R $$

$$ \phi' = \arctan\left(\frac{ẏ_C}{ẋ_C}\right) \qquad s' = \sqrt{(ẋ_C)^2 + (ẏ_C)^2} $$

Numerical solutions for the centroid velocity of the ball during compression and restitution phases.

$$ (\dot{v_x})_{n+1} - (\dot{v_x})_n = - \frac{1}{M} \left[\mu_w \cos(\phi) + \mu_s \cos(\phi') \cdot (\sin \theta + \mu_w \sin(\phi) \cos \theta)\right] \Delta P_I $$

$$ (\dot{v_y})_{n+1} - (\dot{v_y})_n = - \frac{1}{M} \left[ \cos \theta - \mu_w \sin \theta \sin \phi + \mu_s \sin \phi' \cdot \left( \sin \theta + \mu_w \sin \phi \cos \theta \right) \right] \Delta P_I $$

Numerical solutions for angular velocity of ball

$$ (\dot{\omega_x})_{n+1}−(\dot{\omega_x})_n = -\frac{5}{2MR}[\mu_w \sin(\phi) + \mu_s \sin(\phi') \times (\sin(\theta) + \mu_w \sin(\phi)\cos(\theta))]\Delta P_I $$

$$ (\dot{\omega_y})_{n+1}−(\dot{\omega_y})_n = -\frac{5}{2MR}[\mu_w \cos(\phi)\sin(\theta) - \mu_s \cos(\phi') \times (\sin(\theta) + \mu_w \sin(\phi)\cos(\theta))]\Delta P_I $$

$$ (\dot{\omega_z})_{n+1}−(\dot{\omega_z})_n = \frac{5}{2MR}(\mu_w \cos(\phi)\cos(\theta))\Delta P_I $$

$\theta$ is a constant of the angle of cushion contact above ball centre with $\sin(\theta) = 2/5$. $\mu_s$ is the coefficient of sliding friction between the ball and table surface. $\mu_w$ is the coefficient of sliding friction between the ball and the cushion.

Work done by the normal force at contact point $I$ along the $Z'$-axis which is aligned from the ball centre to I

$$ W_{Z'}^I(P_I^{(n+1)}) = W_{Z'}^I(P_I^{(n)}) + \frac{\Delta P_I}{2} \left( z'_I(P_I^{(n+1)}) + z'_I(P_I^{(n)}) \right) $$

The ball is assumed to be bouncing in the +y cushion. Compression phase iterates until

$$\dot{v}_y \le 0$$

For the restitution phase the iteration continues until the work done is

$$W_{Z'}^I \ge e_e^2 W_{\text{compression}}$$

Some of the Mathavan equations not supplied by the paper were inferred to bridge gaps for a complete numerical solution.

Based on the book Impact Mechanics by Stronge. This analytical model accounts for the compliant nature of cushion deformation and resolves the collision across three slip regimes.

Contact velocity at the cushion contact point:

$$\vec{V}_c = \vec{v} - (\vec{\omega} \times R \hat{n})$$

Regime classification is determined by the ratio $v_{\text{ratio}} = v_{t0} / v_{n0}$ and thresholds involving the friction coefficient $\mu$, restitution $e_n$, and mass-matrix coefficients $\beta$:

where $\eta^2$ is derived from the frequency ratio $\omega_t/\omega_n$.

Velocity reconstruction from scalar solver results $v_{nf}$ and $v_{tf}$:

$$\Delta v_n = \frac{v_{nf} - v_{n0}}{\beta_n} , \quad \Delta v_t = \frac{v_{tf} - v_{t0}}{\beta_t}$$

Final updates:

$$\vec{v} \leftarrow \vec{v} + \Delta v_n \hat{n} + \Delta v_t \hat{t}$$ $$\vec{\omega} \leftarrow \vec{\omega} + \frac{mR}{I} (-\hat{n} \times \Delta v_t \hat{t})$$

nvm use v24.11.0
corepack enable
yarn set version 4.9.1
yarn install
yarn build
yarn gltfpack

This generates artefacts in /dist for prod deployment (e.g. on github static pages)

yarn serve

Then open http://localhost:8080/ in your browser to play

yarn test
yarn coverage

yarn deps
yarn upgrade -L
yarn prettify

yarn serve

then open http://localhost:8080/multi.html to see options, message server is public nchan.

Use mouse, trackpad, touch screen or keyboard:

⇦ ⇨ Aim

Control ⇦ ⇨ Fine aim

⇧ ⇩ Topspin and backspin

Shift ⇦ ⇨ Side spin

Space Hit - hold for more power

A Toggle aim helper

M Masse angle

O Camera view

F Full screen

C Chat

H Help

+ - Camera height

scroll Shot power

dbl click Hit

The physics engine is lightweight and deterministic, making batch rollouts, parameter fitting, and simulation experiments practical:

• Throughput & batch execution: Performs ~500 rollouts/sec on 4 CPU cores in pure TypeScript, scaling across worker threads. Runs in-browser via Web Workers or headlessly via Node.js for parallel batch runs.
• System identification & real-to-sim calibration: Designed for trajectory fitting and physics calibration against recorded real-world shot trajectories. Used to estimate physical parameters such as rolling friction, sliding friction, spin decay, restitution, and cushion deflection.
• Reinforcement learning & planning: Suitable as a fast test environment for RL agents, continuous control, Monte Carlo Tree Search (MCTS), trajectory forecasting, or synthetic dataset generation.
• Parameter optimization: The multi-shot physics optimiser provides a starting point for fitting parameters and exploring the loss landscape against recorded shots.
• Worker interface & headless runs: The Web Worker / simulation research page documents the worker protocol, parallel rollouts, headless Node.js scripts, batch processing, and runtime parameter overrides.

Contributions and experiments are welcome.

July 2018

July 2019

March 2021

August 2023 (mobile)

top	aim

Star History

This project is open source and licensed under the GNU General Public License - see the LICENSE file for details. Contributions welcome.