Regis Petit's Website

Physics of Billiards

Contents of this page

1. Physics of Billiards
2. Websites
   1. Articles and general studies
   2. Carom Billiard simulator
   3. Video
3. Another ball games
   1. Bowling
   2. Golf or football

1. Physics of Billiards

2. Websites

2.1. Articles and general studies

1. Régis Petit, Bibliographie
2. Wikipedia, Billard
3. Wikipedia, Carom Billiard
4. D.G. Alciatore, Pool and Billiards Physics Resources
5. Articles from Billiards Digest by Bob Jewett
6. J. Walker, The Physics of the Follow, the Draw, and the Massé, Scientific American, July, 1983 (translated in "Pour la science", September 1983) (.pdf, 600 Ko)
7. Edmond aide mon billard - La référence pour tous joueurs de billard
8. Mathieu Bouville : Sciences appliquées au billard
9. Kozoom, Site spécialisé dans l'édition de produits axés sur le billard
10. FFB : Fédération Française de Billard

2.2. Carom Billiard simulator

1. "Caromball" de Jean-Michel Fray (équations de Coriolis, modifiées par J.M. Fray)
      Présentation de Caromball (FR) (YouTube, 01:57)
2. "Coriolis3D" by Jean-Luc Frantz and Guy Grasland (R. Petit's vector equations modified by J.L. Frantz concerning impact against the cushion)
3.    R. Petit's vector equations (included in many Billiard simulators)
4. "VRCarom" de VRBillard (équations de Coriolis reprises par J.A. Gomez Valderrama)
5. "Billard Carambole 2D" by Alain Delawoevre (Coriolis's equations modified by A. Delawoevre)
6. Simulator by Laurent Buchard (personal equations)
7. "Simulateur de billard réaliste" par Julien Ploquin - Thèse de doctorat

2.3. Video

1. Le Billard. Français, Américain, Snooker et Pool (YouTube, 01:00)
2. Virtual Pool 3 (YouTube, 01:13)
3. crazy billiards (YouTube, 01:20)
4. 3-dimensional billiards (YouTube, 01:32)
5. BILLARD SANS TRUCAGE (YouTube, 01:34)
6. Semih Sayginer "One Man Show" (YouTube, 01:44)
7. HSV B.3 - various jump shot techniques (YouTube, 01:45)
8. Carom 3D shots (YouTube, 02:45)
9. Billard Carambole OpenGL C++ (YouTube, 03:15)
10. pool and billiards instructional video excerpts (YouTube, 03:41)
11. Venom Trickshots II- Episode III: Sexy Pool Trick Shots in Germany (HD) (YouTube, 04:45)
12. Billard Artistique (YouTube, 06:52)
13. Eric Yow! and Mike Massey, Artistic Billiards (YouTube, 06:54)
14. Artistique-Billard Figuren Satz H (YouTube, 09:45)

3. Another ball games

3.1. Bowling

Physics of bowling is similar to physics of billiards. The cloth corresponds to the bowling track, and the dry friction between ball and cloth becomes the viscous friction between ball and oiled track.
The equations of ball motion are identical to those of billards, except the force (F) and the vertical torque (K vertical) which become :
F = M g (z - fv WE)
K vertical = - (kz / R) (M g R) Ω vertical
fv = coefficient of viscous friction between ball and track (in s/m)
kz = coefficient of boring viscous friction between ball and track (in s.m)

The vector resolution gives the following results :
W = Wi - (1 - exp[ -j t ]) (2/7) WEi
Ω horizontal = Ωi horizontal - (5/ 2R) z x (W - Wi)
Ω vertical = exp[ -k t ] Ωi vertical
F = M g (z - fv exp[ -j t ] WEi)
WEi = Wi + R z x Ωi
j = (7/2) fv g
k = (5/2) (1/ R²) kz g
The ball slides on the oiled track and never reaches the rolling without slipping.
The trajectory (OG) of the center of gravity G is given by the following equation :
OG = OGi + t Wi - (t + (1/j) exp[ -j t ] - (1/j)) (2/7) WEi
Nota : This trajectory is not a parabola as billiards.
The rotation angles (A) of the ball around the G center are given by the following equations :
A horizontal = Ai horizontal + t Ωi horizontal + (5/ 7R) (1/j) (j t + exp[ -j t ] - 1) z x WEi
A vertical = Ai vertical + (1/k) (1 - exp[ -k t ])) Ωi vertical

3.2. Golf or football

Physics of golf or football is more complex than physics of billiards. The cloth corresponds to the air surrounding the ball, and the dry friction between ball and cloth becomes a sum of three factors : resistance to movement (drag Tr), resistance to own rotation (resistant torque Cr) and Magnus effect due to the own rotation (lift Po).
The high velocity of ball through air gives a turbulent flow. The equations of ball motion including the wind speed become :
M dW/dt = F - M g z
I dΩ/dt = Cr
F = Tr + Po
Tr = - kr ||W - W wind|| (W - W wind)
kr = (1/2) Cx (r air) π R²
Cx = drag coefficient (0.14 for a sphere)
r air = density of air
Po = kp Ω x (W - W wind)
kp = lift coefficient
Cr = - kc Ω
kc = viscous friction coefficient

Vector resolution is complex but reachable under some simplifying conditions.

Copyright © 2005 Régis Petit.               Last page update : December 29, 2023.