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 - MassÃ© 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

TP B.22 - How peak tip contact force and contact patch size vary with shot speed, and drop tests

**See also:**

High-speed pool and billiards video clips

Normal-speed pool and billiards video clips

Instructional articles