Comparison of bisector-point and double-angle-bisect draw systems

technical proof
technical proof
TP A.21
Comparison of bisector-point and double-angle-bisect draw systems
cut
angle
( φ)
OB
CB
GB
original
line of
centers
P
θ
aiming
line
supporting:
“The Illustrated Principles of Pool and Billiards”
http://billiards.colostate.edu
by David G. Alciatore, PhD, PE ("Dr. Dave")
originally posted: 11/29/05 last revision: 11/29/05
change in
cue ball
angle
( θc) θ
bisector
final cue
ball direction
ball radius: R := 1.125⋅in 2R
distance
between
CB and OB
O
θ
d
C
φ
θ+β
β
G
P
φ+β
R
ghost ball
center
Note - the change in cue ball angle θ c (e.g., see TP A.20) is related to the bisector angle θ according to:
θc =
180⋅deg − 2⋅θ − β

From the triangle formed by C, P, and the perpendicular from P to OC,
tan( β)
βθd ,
( )
( )
R⋅sin θ
=
d − R⋅cos θ
( ) := atan2( d − R⋅cos( θ)
, R⋅sin( θ)
)
Applying the law of sines to triangle OGP gives:
⎡⎣ ( ) ⎤⎦
sin 180⋅deg − θ + β
φθ, d
2R ⋅
( ) ⎛
:= asin⎜ ⎝
=
sin θ β θ d , +
sin( φ + β)
R
( ( ) ) ⎞
2 ⎠
−
βθd ( , )
The double-angle-bisect system (see Bob Jewett's October, 1995 BD article) predicts:
φ ( dab θ)
θ
:=
2
Here's how the methods compare for two different CB-OB distances:
d1 := 1ft ⋅
d2 := 6ft ⋅
θ := 0⋅deg .. 60⋅deg φ
dab
( θ)
deg
( )
φθd ,
1
deg
( )
φθd ,
2
deg
30
20
10
0
0 20 40 60
So the two systems agree fairly well for small cut angle shots. There is more disagrement
at bigger cut angles, especially when the distance between the CB and OB is large.
θ
deg