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*

*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.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.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 during a break shot

**See also:**