A Mathematician Catches a Baseball
EdwardAboufadel
1. INTRODUCTION.In the game of baseball, what strategydoes an outfielder
employ to catch a fly ball? Recently, Michael McBeath and Dennis Shaffer,who
are psychologists,and MaryKaiser,a researcherat NASA, proposeda new model
to explain how this task is accomplished[1]. The model, called the linear optical
trajectoty (LOT) model, was developed and tested empirically by the three
researchers,and it receivednationalattentionduringthe 1995 baseballseason [4].
In this paper, seekingto clarifywhat is writtenin [1] and [4],we develop equations
relatingthe motion of a fly ball to the motion of an outfielder utilizingthe LOT
strategy.In the process,we providea mathematicalfoundationon which the LOT
model can rest.
To begin, let H be home plate, B the positionof the ball, and F the positionof
the fielder at any point in time; see Figure 1. Define B* to be the projectionof B
onto the playingfield, so that H, F, and B* are co-planar.There is a well-defined
point I*, which is the point of intersection of the line B*F and the unique
perpendicularto B*F throughH. There is anotherwell-definedpoint I, which lies
on the line BF, directlyabove I*. The point I is the fielder'simage of the ball.
/
t
/tN
\
Figure
1
There are three important angles defined in the right pyramid HFII*: the
verticaloptical angle a = Z B*FB, the horizontaloptical angle ,l]= Z B*FH, and
the optical trajectoryprojectionangle t = X I*HI. We then have:
The LOT model hypothesis. The strategythat a fielderuses to catch a flyball is to
follow a path that will keep the opticaltrajectozy
projectionangle t constant;this is
equivalentto keepingthe ratio(tan a)/(tan ,B) constant.
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A MATHEMATICIANCATCHESA BASEBALL
[December
In discussingthe LOT model, McBeath and his colleagues write, "The LOT
strategydiscernsoptical accelerationas optical curvature,a feature that observers
are very good at discriminating,"
and, "If you'rerunningalong a path that doesn't
allow the ball to curve down, then in a sense you are guaranteedto catch it."The
LOT model apparentlyalso applies to other situations, such as the pursuit of
mates and prey by certain fish and houseflies. McBeath et al. also write, "[The
LOT model] keeps the image of the ball continuouslyascendingin a straightline
throughout the trajectory."[1,2] This last statement can be misleading to the
casual reader who assumes it means that the trajectoryof I must be linear. In
Section 4, we clarify this statement and others that have been made about the
model.
2. THE FIELDER,HIS PREY,AND THE IMAGEOF HIS PREY. If a fielder uses
the LOT model, is his path uniquelydeterminedby the path of the baseball? In
this section,we develop equationsrelatingthe positionsof the fielder, the ball, and
the image of the ball, as we seek an answer to this question. Our analysis is in
three-dimensionalspace, with the playing field represented by the xy-plane.
Without loss of generality,let home plate H be the origin, and identify the first
and thirdbase lines with the x-axisand the y-axis,respectively,so that a fair ball is
one that lands in the first quadrantof the plane. The coordinatesof our three
relevantpoints are F = (Xf, yf, Zf), B = (xb,Yb, Zb) and I = (xi, Yi,zi) All nine
coordinatesare functionsof time t (with t = 0 representingthe moment that the
ball is hit by the batter), and Zf= 0 at all points in time.
We define two other functionsof time:
p =-
and q =-.
( 1)
If the trajectoryof I is linear, then p and q would be constant. However,in the
LOT model, this is not necessary.Instead,we have the followinglemma:
Lxmma2.1. If theLOTmodelis valid(i.e., if
constant.
t
is constant),thenq2/(l
+ p2)
is
Proofof Lemma2.1: If we considerFigure 1, we see that
ls*l2
z2
q2
ta-n2t = lHI*l2 = x2 +y2
1 +p2
(2)
It is helpful to first consider the case where p and q are constant, so we also
introduce:
Tlle strongLOTmodel hypothesis. Thestrategy
thatthefielderusesto catcha flyball
is tofollowa paththatkeepsbothp andq constant.
For either hypothesis,the line HI* has slope p, so it follows that the line B*F
has slope -1/p; see Figure 2. Therefore,using the definitionof slope, we have
_ _
p
=
Yf
Xt-Xb
Yb
Xb -Xt
Yf
Yb
and (3) is true at everypoint in time.
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A MATHEMATICIANCATCHESA BASEBALL
871
whent = T. Weassume that
PYb0 as t T.
Xb +
F*
lgllre w
_
Next, we determine xi in terms of B and p. The equation of the line HI* is
y = px, and the equationof the line B*F is y = Yb-(x-Xb)/p.
The point I* is
determinedby the intersectionof these two lines, and by setting these equations
equal to each other and solving,we get
Xb + PYb
p2
+
1
*
(4)
Since F, B, and I are collinear, we have (zi-Zf)/(Xi-Xt)
= (Zb-df)/
(Xb - Xt), and since Zf = 0 and zi = qxi, solvingthis equation
for Xt gives
Xt
Xi(
Z
-qX
)-
(S)
Combining(4) and (5) and simplifyingyields
(Zb - qXb)(Xb + PYb)
Zb(P + 1) q(xb + PYb)
Then, combining(3) and (6) and simplifyingleads to
( PZb
t
Zb(p2
4!Yb)(Xb
+
+
1) -q(xb
(6)
PYb)
(7
+<Yb)
Finally,if we solve (6) for q, we get
{
Zb
qXb +
8
PYb
Xt(p2
+
1) -
(Xb
+
PYb)
.
Xt-Xb
J
(8)
What we now have, for every t > Oand for every trajectoryB, is a relationship
betieen (Xt, yt) and (p, q). If we knowp and q, then (6) and (7) yields Xt and yt,
and if we know Xt and yt, then (3) and (8) give us p and q. In the next two
sections,we use these relationshipsto investigatethe subtle elements of the LOT
model.
3. THE STRONGLOT MODELHYI?OTHESIS:
CONSEQUENCESAND PROBLEMS. For a given ball trajectoryB, let T be the time when the ball is either
caughtor hits the ground, so that the ball is in flight for 0 < t < T and Zb = °
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AMATHEMATICIANCATCHESA BASEBALL
[December
Theorem3.1* For a given ball trajectozyB, and for euezyto such that 0 < to < T,
thereexistsa uniquefielder'spath, dependingon thepositionof thefielderat time toS
such that thefieldercan use the strongLOT modelfor to < t < T on thatpath and
catchthe ball at time T.
Proof of Theorem3.1: At time to the positions of the fielder and the ball are
known.Therefore, using (3) and (8), we can determine p and q. Once these two
constants are known,(6) and (7) specifies the unique path the fielder follows. At
time T, we have Zb = °, and therefore
(Zb
f
zbt p2
Similarly,yf It=
T =
-qXb)(Xb
+
qxb(xbPYb)
q(Xb + PYb)t=T
+PYb)
1) -q(Xb
+
PYb)t=T
b
Yblt=t *
Thus, the strongLOT model works,provided,of course,that the fielder can run
fast enough to follow his predeterminedpath. Can a fielder tracka ball startingat
the momentthe ball is launchedby the batter?McBeathand his colleagueswrite,
" . . . fielders do not and cannot arbitrarilyselect optical angles and rates of
change. . . but ratherthey maintainthe initial optical projectionangle, , which is
fully determinedby the perspectivelaunchangle of the ball relativeto the fielder."
[2] This suggeststhat formulascan be developed for p and q, and hence , from
the informationat t - 0, and that perhapsthe strongLOT strategycan be utilized
from the moment the ball is hit.
I*mma 3.1. Underthe assumptionthatp and q are constantneart = 0, the valuesof
p and q are uniquelydetemxined
at t = Oby the initialuelocityof the baXand the initial
positionof thefielder.
Proofof Lemma3.1: Since (3) is true for all t, it is true for when the batterhits the
baseball(t = 0), thereforewe can conclude that
Pl
Xf
= Xb
= _ Xf
(10)
To determine q, we use (8) and L'Hopital'srule:
ql,=o= lim
f
tO
= lim (
Xb
Zb \
+ PYb
Xf(p
)(p2
+
1)
(Xb
+
PYb)
Xt-Xb
+
1) =
(
/
)
(ll)
Thus,(10) and (11) provideformulasfor p and q as functionsof the positionof the
fielder and the velocityvectorof the ball at t = 0.
X
Lemma 3.1 clearly applies to the strong LOT model, but, as we see in the
followingtheorem,we now have a problemwith the strongLOT strategyat t = 0,
because q does not depend on the initial position of the fielder.
Theorem3.2. For a givenball trajectozy
B, thereexistpoints (Xt, yt, O), called ideal
fielders'positions, such thata fieldersituatedat thatpositionwhenthe ball is hit can
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A MATHEMATICIANCATCHESA BASEBALL
873
use the strongLOT modelfor 0 < t < T to detenninea uniquepath to catchthe balZ.
Not all fielders'positionsare idealfielders'positions.
Proof of Theorem3.2:
Choose
q as a function
determine
an arbitrary
value
of p. From
Lemma
3.1, we can
of B and p and define
Xt
t0
_
1
( Zb - qxb ) ( Xb
Zb( p2 + 1) -q(xb
PYb )
+ PYb)
(12)
and
( 13)
) ( Xb+ PYb)
( PZb qYb
situated at (Xt, yt, O)at t = 0 can then use the unique path described by
(6) and (7) in order to catch the ball at time T.
at (Axt, Ayf,0) when
a second fielder situated
Now fix A + 1 and consider
t = 0. This fielder has the same p and q as the first fielder (due to Lemma 3.1),
The
the same running path as the first fielder, which is impossible.
and therefore
f
second fielder is not in an ideal fielder's position.
A fielder
on the velocity vector of the ball, a
matter, since q depends
As a practical
fielder would need some period of time after the ball was hit to recognize what t
is. This means an ideal fielder is ideal in another sense, because at t = 0, he can
discern the velocity of the baseball and begin his path to catch the
instantaneously
of how
3.1, which doesn't apply when t = 0, is a better description
ball. Theorem
the strong LOT model operates, and it is simply incorrect to say that a fielder can
use the LOT strategy from the crack of the bat. It makes more sense to say that
a moment after the ball is
,
which is determined
fielders maintain a constant
launched
by the batter.
TO CLARIFY IDEAS. In the
4. ONE PATH OR MANY: USING MATHEMATICS
case where we use the LOT model and not the strong LOT model, there is no
longer a unique path that a fielder must follow in order to achieve a linear optical
trajectory.
B, and for evezyto such that 0 < to < T,
4.1. For a given ball trajectozy
thereexistan infinitenumberof fielders'paths, such that a fieldercan use the LOT
strategyfor to < t < Ton thatpathand catchthe ball at time T.
Theorem
Proofof Theorem4.1: This theorem
both p and q
now, Es t increases,
q2/(l
+ p2)
remains
constant.
is proved the same way as Theorem 3.1, except
are allowed to vaxy, as long as, by Lemma 2.1,
q
This gives an infinite number of solutions.
of Lemma 3.1, we can also define ideal fielders' positions
With the assumption
4.2, except that now the
for the LOT model. The proof is the same as Theorem
by (12) and (13) is not unique.
path determined
We can use our results to examine the validity of, and explain, several stateanalysis is that we can
ments from [1] and [2]. The benefit of the mathematical
follow from the LOT model and gain a greater
how these statements
recognize
874
A MAIsHEMATICIANCATCHES A BASEBALL
D.
cemter
appreciationfor the model. Here are the statements:
A.... the [LOTmodel]strategyin itself does not specifya uniquesolution.[2]
B. ... angle of bearingappearsto be used as an additionalconstraintto help determinethe
particularLOT chosen. [2]
C. "[The LOT model] keeps the image of the ball continuouslyascendingin a straightline
throughoutthe trajectoly."[1]
D. One interestingaspect that has emergedirom researchon this problemis that for identical
launches,fielderswill select differentrunningpaths,particularlynear the beginningand end of
the task. A good model of outElelderbehaviorshould allow for this variability,as the LOT
strategy does. Near the beginningof the trajectory[of the ball], we expect more variability
because outfielderlocationhas less influenceon the opticaltrajectory.Near the end we expect
morevariabilitybecausecorrectiveactionwill commenceas other depth cues become available.
[2]
StatementA is a result of Theorem4.1. The angle of bearingin statementB is a
functionof p and q, and since q is a functionof p by Lemma2.1, it makes sense
to call p the fielder'sbeanngfunction. As the fielder tracks the ball, he unconsciously chooses a function p as a part of his LOT strategy. (The strong LOT
model keeps the angle of bearingconstant.)Since it is reasonableto assumethat a
player'sbearingswill not change in the first moment after the ball is hit, we are
justified in keeping p constantnear t = 0 in Lemma3.1.
StatementC seems at odds with the idea that p and q can be allowed to vary,
which suggests that the LOT hypothesisdoes not allow the trajectoryof I to be
non-linear.A fielder's bearings can change, and this correspondsto a rotation
about home plate of the right pyramidin Figure 1. For example, if the bearing
functionchangesfrom a value of P1 to a value of P2, then the angle of rotationis
0 = arctanP2 - arctanP1- From the rotationof axes formulas,we get
COS0 =
and sin 0 =
1 +P1P2
\/(1 + p2) \/(1 +P2)
P2-P1
. (15)
/(1 + p2) 1/(1 +P2)
Suppose we have a situationwhere first we have I1 = (xi, p1xi, q1xi) and then a
change of bearings leading to I2 = (Xis p2Xi, q2Xi). Applyingthe change of variables formulato I2, we end up with the rotated point
1 +P2
1 +p2 (xi,p1xi,qlxi).
I2 =
(16)
From the perspective of the fielder, the ball appears to remain on the vector
(1, P1, q1>, "continuouslyascendingon a straightline."
StatementD is a consequenceof the proof of Theorem4.1, which explainshow
there can be manypaths that keep t constant.Also, from (3), we get
dp
dXt
=-
1
Yf
Yb
and
dp
- = -
dYf
Xb-Xt
( Yf
Yb )
2.
(17)
These derivativesare small when t is near 0, as are the derivativesfor q. As a
consequence,when (3) and (8) are used to determinep and q "near the beginning
of the trajectory,"fielders near each other have quite similar optical trajectory
projection angles , hence "outfielder location has less influence on optical
trajectory."Therefore, differentrunningpaths can keep p and q nearlyconstant,
as the LOT model predicts.
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A MATHEMATICLANCATCHESA BASEBALL
875
fielder chooses a path such that
Xb +
PYb0 as t T. In this case, Theorems
3.1
The mathematicspresented here shows that the LOT model is reasonableand,
interestingly,qualitativeobservationsmade by the researcherscan be supported
quantitativelyby the analysis.This analysis,though, doesnotprovethat the LOT
model is correct.The LOT model was developedby perceptualpsychologistsusing
statisticalmethodsand it cannot be provedas we prove a theoremin mathematics.
It also goes without saying that outfielders do not and cannot follow the LOT
strategyexactZy, or even that outfieldersfollow the strategywell. In [1], McBeathet
al. reportedthat one fielder appearedto use a linear optical trajectoxyfor a while,
faltered, then continuedon a new optical trajectorywith a different t!
There are other, competingmodels,such as the optical accelerationcancellation
(OAC) model, that have their defenders. Accordingto the OAC model, a fielder
acts to keep d(tan a)/dt constant, not t. Another view comes from Robert K.
Adair, a physicist,who argues that 4'afielder runs laterallyso that the ball goes
straightup and down from his or her view."[3] The OAC model or Adair'smaybe
correct, although McBeath and his colleagues rebutted both theories with this
statement, "Both maintenance of lateral alignment and monitoringof up and
down ball motion require informationthat is not perceptuallyavailablefrom the
fielder'svantage."[2]
An exampleof the failureof the LOT strategywas recentlypresentedby James
L. Dannemiller,Timothy G. Babler, and Brian L. Babler. They write that it is
possible for a fielder to use the LOT strategyand arrive 4'awayfrom the ball's
landing site at the instant the ball hits the ground."[5] This is possible if the
and 4.1 are invalidand B + F when t = T. However,in this instance,H, B, and F
would become collinearat the moment the ball hits the ground.
5. A MATHEMATICIAN
CATCHESA BASEBALL.We now go to the ball park,
and a mathematicianon the visitingteam is standingin rightfield, waitingto catch
some fly balls for his team. It is the bottom of the ninth inning,and the visitorsare
ahead 4-3. The ball is hit! Let's supposethat B*, the projectionof the ball on the
field, moves in a straightline and that the path of the ball is a parabolicarc. These
are reasonable assumptions(unless we are playing in WrigleyField, where it is
ratherwindy at times) and an exampleof a ball?strajectory(in feet) would be
xb(t) = 75t,
Yb(t) = 10t,
zb(t) = -64t2 + 256t.
(18)
This is a ball hit to deep right field that will be in the air for 4 seconds and will
land approximately303 feet from home plate unless our intrepid mathematician
gets there in time.
Suppose that our right fielder is positioned in right-centerfield at the crackof
the bat at the position(Xt, yt, Zf ) = (270, 70, 0), that it takes him 0.3 seconds to get
his bearings,and that he plans to use the strong LOT model to catch the ball.
UtilizingTheorem 3.1, he determinesthat
p
-3.694
and
q
99.121.
(19)
Therefore,t
87.8°;this is practicallyvertical perhapsAdair has a point! Now
that p and q are known,a uniquepath can be determinedfor our all-starto catch
this fly ball. That path is indicatedin Figure 3.
On the very next play, inexplicablyanother ball is hit to right field with the
same trajectoryand the same response time for our athlete. This time, though?he
decides to use the regularLOT strategy?computinghis initial p and q as above,
and then using p = -0.0827t- 3.6692 as his bearing function. This path?which
876
A MATHEMATICIANCATCHESA BASEBALL
[December
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+
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<)°
C
tC
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20
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250
+
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260
0
§
' I ' ' ' ' I { f
270
280 x
t
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290
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t
i,
300
t
+
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i
310
t
t
f
{
{
+ §
320
Figure 3
sends the fielder to the edge of the playing field and back, is also indicated in
Figure 3.
Now that there are two outs, our mathematicianwondersif the next batterwill
also hit a fly ball satisfying(18). He decides to get into an ideal fielder'sposition.
Althoughthe most obviousone is (300,40, 0)-the point where the ball has landed
the first two times he decides to find an ideal position that correspondsto his
currentposition.He computesp -3.857, using(10), and q 111.579,using(8).
Then, makinguse of equations(12) and (13), and a little calculus, he determines
Xt
291 and yt
75.
(20)
Figure 4 shows, for a ball following trajectoty (18), several ideal fielder's
positions, depending on the choice of p. Again in Figure 3, we see out hero,
1996]
A MATHEMATICIANCATCHESA BASEBALL
877
300 -
250 -
200 t
+
t
t
i
,+,
Y
+
150
+
+
100-
+
+
so
'
0
+
,
O
,
f ,
50
,
t
'
8
+
,
,
,
100
+
t
150
,
,
I ,
200
+
,
I
250
0
,
*
E
t
300
x
Figure 4
startingat (20), pursuingthe ball using the strong LOT strategyand catching it
after 4 seconds. That'sthree outs game over!
REFERENCES
[1] M. McBeath,D. Shaffer,and M. Kaiser,How BaseballOutfieldersDetermineWhere to Run to
CatchFly Balls, Science268 (28 April 1995),569-573.
[2] M. McBeath,D. Shaffer,and M. Kaiser,Letters,Science268 (23 June 1995),1683-1685.
[3] <. Adair,Letters,Sciences268 (23 June 1995),1681-1682.
[4] P. Hilts, New TheoryOfferedon How OutfieldersSnagTheir Prey,TheNew YorkTimes,28 April
1995,B9.
[5] J. Dannemiller,T. Babler, and B. Babler, On CatchingFly Balls, Science 273 (12 July 1996),
256-257.
Departmentof Mathematics& Statistics
GrandValleyState University
Allendale,MI 49401-9403
aboufade@gvsu.edu
878
A MATHEMATICIANCATCHESA BASEBALL
[December