teacherteacherTechnical Proof Pool and Billiards Analyses

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The technical proofs below contain physics background and mathematical formulas to support many of the principles in the book. Refer to the book to see how the principles are used in practice.

NOTE: this information will be of interest only to people with strong physics and mathematics backgrounds. Others proceed at your own risk.


Chapter 2 - Fundamentals

TP 2.1 - Minimum cue tip friction required for no-slip horizontal impact

Chapter 3 - Executing Basic Shots

TP 3.1 - 90° rule
TP 3.2 - Ball speeds and distances after stun-shot impact
TP 3.3 - 30° rule
TP 3.4 - Margin of error based on distance and cut angle
TP 3.5 - Effective target sizes for slow shots into a side pocket at different angles
TP 3.6 - Effective target sizes for slow shots into a corner pocket at different angles
TP 3.7 - Effective target sizes for fast shots into a side pocket at different angles
TP 3.8 - Effective target sizes for fast shots into a corner pocket at different angles
TP 3.9 - Pocket target center offsets based on angle to the pocket (see TP 3.5 through TP 3.8
TP 3.10 - Pocket margin of error regions for a slow shot (see TP 3.5 and TP 3.6
TP 3.11 - Pocket margin of error regions for a fast shot (see TP 3.7 and TP 3.8
TP 3.12 - Object ball angle error based on cut angle (see TP 3.4

Chapter 4 - Spin and English

TP 4.1 - Distance required for stun and normal roll to develop
TP 4.2 - Center of percussion of the cue ball
TP 4.3 - Spin-induced throw effects
TP 4.4 - Relationship between the amount of throw and cut angle

Chapter 6 - Bank and Kick Shots

TP 6.1 - Parallel midpoint-line banking method geometry
TP 6.2 - Mirror-image kick-shot method geometry
TP 6.3 - Increase in bank rebound angle due to the rail coefficient of restitution

Chapter 7 - Advanced Techniques (Shot Making)

TP 7.1 - Two-rail parallel-line kick-shot geometry
TP 7.2 - Corner-5 three-rail diamond system formulas
TP 7.3 - Ball-rail interaction and the effects on vertical-plane spin


Additional proofs not referenced in the book (but referenced in Billiards Digest articles):

TP A.1 - Half-ball hit, foot-spot carom shot
TP A.2 - Carom shot vs. cut shot example

TP A.3 - Minimum cue elevation required for a head-spot-to-foot-spot center-ball-hit shot

TP A.4 - Post-impact cue ball trajectory for any cut angle, speed, and spin
TP A.5 - The effects of ball inelasticity and friction on the 90° rule
TP A.6 - The effects of ball inelasticity and friction on the 30° rule
TP A.7 - The effects of sidespin on the 90° rule
TP A.8 - The effects of sidespin on the 30° rule

TP A.9 - Cue accelerometer measurements
TP A.10 - The effect of bridge length on contact point accuracy
TP A.11 - Analysis of a common fractional-ball aiming system
TP A.12 - The relationship between cue ball spin and cue tip offset
TP A.13 - Number of lines of aim required for different types of shots

TP A.14 - The effects of cut angle, speed, and spin on object ball throw

TP A.15 - Controlling the cue ball direction in a frozen cue-ball shot
TP A.16 - Final ball speeds, distances, and directions for natural roll shots
TP A.17 - The effect of the amount of friction on throw
TP A.18 - Distance required for stun and natural roll to develop for different tip offsets
TP A.19 - Masse shot aiming method, and curved cue ball paths

TP A.20 - The effect of spin, speed, and cut angle on draw shots
TP A.21 - Comparison of bisector-point and double-angle-bisect draw systems
TP A.22 - "Tips" of english
TP A.23 - Ball-hit fraction vs. cut angle

TP A.24 - The effects of follow and draw on throw, and OB swerve
TP A.25 - The relationship between spin-rate-factor and percent-english
TP A.26 - The amount of sidespin required for "gearing" outside english
TP A.27 - Spin transfer
TP A.28 - Throw plots for all types of shots
TP A.29 - Using throw to limit cue ball motion

TP A.30 - The effects of cue tip offset, cue weight, and cue speed on cue ball speed and spin
TP A.31 - The physics of squirt


TP B.1 - Squirt angle, pivot length, and tip size
TP B.2 - Rolling resistance, spin resistance, and "ball turn"
TP B.3 - Throw calibration and contour plots for various cut angles, speeds, English, and roll
TP B.4 - Stroke speed and acceleration vs. distance
TP B.5 - Rolling CB, direct-hit hop and ball travel distances
TP B.6 - Cue ball table lengths of travel for different speeds, accounting for rail rebound and drag losses
TP B.7 - Effect of squirt on the amount of spin
TP B.8 - Draw shot physics
TP B.9 - Draw shot spin vs. spin ratio
TP B.10 - Draw shot cue elevation effects
TP B.11 - Shallow-angle contact-point mirror kick system
TP B.12 - Optimal tip height for speed/distance control
TP B.13 - Rolling cue ball deflection angle approximations
TP B.14 - Draw shot cue ball angle approximations
TP B.15 - Pocket geometry calculations
TP B.16 - CB-OB distance for follow cling
TP B.17 - Maximum Drag-Enhanced Sidespin Tip Contact Point
TP B.18 - Pendulum Stroke Cue Tip Trajectory
TP B.19 - Comparison of cue ball deflection (squirt) "endmass" and stiffness effects
TP B.20 - Peak forces and tip contact distance during a break shot
TP B.21 - Small-gap-combination throw effects


See also:

High-speed pool and billiards video clips
Normal-speed pool and billiards video clips
Instructional articles


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