Catcher of the Fly
I.
II.
III.
IV.
V.
VI.
VII.
Introduction
Statement of Problem
Strong LOT Model
7th Inning Stretch
Unanswered Questions
Competing Models
Conclusion
Introduction
Janel Krenz
Favorite Baseball Team:
Milwaukee Brewers
Ivan Lau
Favorite Baseball Team:
New York Yankees
Lori Naiberg
Favorite Baseball Team:
Chicago Cubs
Ben Rahn
Favorite Baseball Team:
UW-Stout Blue Devils
Chad Seichter
Favorite Baseball Team:
Oakland Athletics
Our group has extensively studied the
paper: “A Mathematician Catches a
Baseball” by Edward Aboufadel.
The paper discusses how an
outfielder catches a fly ball.
We will go on to discuss our findings
on the mathematics of how an
outfielder catches a fly ball.
Statement of
Problem
Background
• Old Theory
– Complex calculations
– Solved the problem in 3 dimensions
– Fielders run straight path
• New Theory
– Linear Optical ball Trajectory (LOT)
– Outfielder uses a curved running path
– Fielder keeps the ball on a straight line
Aboufadel’s Mathematical Model
H = Home Plate
B = Position of Ball
F = Position of Fielder
B*= Projection of Ball onto Field
I = Fielders Image of the Ball
I* = Unique Perpendicular
• Aboufadel derived the Strong LOT Model,
which is a special case of the LOT model.
• The Strong LOT Model hypothesis: The
strategy that the fielder uses to catch a fly
ball is to follow a path that keeps both p and
q constant. (p = yi/xi, q = zi/xi)
• With F = (xf, yf, zf), B = (xb, yb, zb), and
I = (xi, yi, zi)
Strong LOT
Model
The Strong LOT Model
Hypothesis
• The strategy a fielder uses to catch a fly ball is to
follow a path that keeps p and q constant
• For this hypothesis, HI* has a slope of p, so it
follows that B*F has a slope of –1/p
xb  x f
1 y f  yb
 
 p
p x f  xb
y f  yb
• Equation (3) is true at every point in time
(3)
• The equation of HI* is y = px and the
equation of B*F is y = yb-(x-xb)/p and the
point I* is determined by the intersection of
these two lines. Set them equal and solve.
( x  xb )
 Subtract y from both
px  yb 

sides and then multiply
p
b
p x  yb p   x  xb 
both sides by p.
p x   x  xb  yb p 
 Add ybp and x to
both sides, factor and
divide and we get
equation (4).
2
2
p x  x  xb  yb p 
2
x ( p 2  1)  xb  yb p
xb  yb p
xi 
(4)
p2 1
• Since F, B, and I are collinear, we have
zi  z f
xi  x f

zb  z f
xb  x f
• And zf = 0 and zi = qxi
zi  0
zb  0


( xi  x f ) ( xb  z f )
(4.5)
qxi
zb


( xi  x f ) ( xb  x f )
 Plug in qxi for zi
and cross multiply.
qxi xb  qxi x f  zb xi  zb x f 
zb x f  qxi x f  zb xi  qxi xb 
x f ( zb  qxi )  zb xi  qxi xb 
 zb  qxb 

x f  xi 
 zb  qxi 
 Factor out,
solve, and we
get equation
(5).
(5)
• Combine equation (4) and (5) to get
equation (6):
xb  pyb  zb  qxb 

 
xf  2
p  1  zb  qxi 
zb  qxb xb  pyb 


 xb  pyb  
 
p  1  zb  q
2
 p 1 

2
xf 

 Substitute
(4) in for xi.

zb  qxb  xb  pyb 
zb  p 2  1  q  xb  pyb 
 Multiply
through and
solve for xf
and we get
equation (6).
.
(6)
• This gives us the x-coordinate of the fielder.
• Now to find the y-coordinate of the fielder
• Solve Equation (3) for y
p
f
xb  x f
y f  yb
py f  pyb  xb  x f 
yf 
xb  x f  pyb
p
 Add pyb to both
sides, divide both
sides by p and we
to get yf.
• Combining yf and equation (6) and solving
we get:
yf

pzb  qyb  xb  pyb 

2
zb  p  1  q xb  pyb 
(7)
• This would give us the y-coordinate of the
fielder.
• Then solving equation 6 for q we get:
2

x
(
p
 1)  ( xb  pyb ) 


zb
f


q  


x

py
x

x
b 
f
b
 b

(8)
• What we now have, for every time t > 0 and for
every trajectory B, is a relationship between (xf,
yf) and (p, q). If we know p and q, we can solve
for the fielder’s xf and yf, and if we know the
fielder’s positions xf and yf then we can solve for
p and q.
•Proof that the fielder will intersect the
ball. (T = time when ball hits ground)
•Using equation (6)
xf
t T
( zb  qxb )( xb  pyb )

zb ( p 2  1)  q ( xb  pyb ) t T
 qxb ( xb  pyb )

 q ( xb  pyb ) t T
 xb
t T
(9)
The same
method
is used to
show that
yf = yb
As a consequence of the Strong LOT
Model, since p and q are constant, you
can calculate them.
Since equation 3 (which
determines the slope of HI*) is
true for all t, it is true when the
batter hits the ball (t=0).
p |t 0 
xb  x f
y f  yb

t 0
 xf
yf
t 0
(10)
To determine q, we use equation 8 and
L’Hopital’s rule.


2

x
p
 1   xb  pyb  
 zb  f


q |t 0  lim 


t 0 x  py
x

x
b 
f
b
 b






 2
zb
z 'b p  1
 lim 
 p 1 
t 0 xb  pyb 
x'b  py 'b
2
(11)
t 0|
th
7
Inning
Stretch
Baseball Trivia
1. Who won last year’s World
Series?
2. What two professional
baseball players broke the
homerun record in 1999?
IT’S PEANUT TIME!!!
ENJOY!!!
Answers:
1. New York
Yankees!
2. Sammy Sosa and
Mark McGwire!
Unanswered
Questions
1. If p and q are not
constant, there is no
unique path.
2. If p and q are not
constant, there might be
a shorter path.
3. If the fielder establishes
the LOT model, can he
run straight to the
destination point?
4. How fast does the fielder
have to be?
Competing
Models
OAC Model
(Optical Acceleration Cancellation)
• Straight running path
• Constant speed
Problems:
• Complex calculations
• This model identifies the projection as a planer
optical projection.
Robert Adair’s Model
• Adair’s Model focuses on the path of a fly ball.
• A fielder runs laterally so that the ball goes
straight up and down from his or her view.
• The lateral alignment and monitoring of up and
down ball motion requires information that is not
perceptually available from the fielder’s vantage.
Conclusion
Wrapping It Up
• Next time you are out on the field, don’t
forget to use the Strong LOT Model!!!
• Remember to keep p and q constant!!!
• Follow these two hints and you will
NEVER miss a fly ball again!!!
Sources
• A. Aboufadel. “A Mathematician Catches a Baseball”. American
Mathematical Monthly. December 1996.
• M. McBeath, D. Shaffer, and M. Kaiser. “How Baseball Outfielders
Determine Where to Run to Catch Fly Balls”. Science.
April 28, 1995.
• P. Hilts. “New Theory Offered on How Outfielders Snag Their Prey”.
The New York Times. April 28, 1995.
• J. Dannemiller, T. Babler, and B. Babler. “On Catching Fly Balls”.
Science. July 12, 1996.