by

D.N. Seppala-HoltzmanSt. Joseph’s College

faculty.sjcny.edu/~holtzman

An Optimal Basketball Free ThrowTo be published in the College Mathematics

Journal of the Mathematical Association of America (Nov. 2012)

Available at: faculty.sjcny.edu/~holtzman downloads

The Free ThrowPlayer stands behind free throw line and, un-

encumbered by opponents and unassisted by team-mates, attempts to throw ball into basket

There are two main components under the player’s control:

The angle of elevation of the shot, The initial velocity of the shot, v0

The ProblemNeither the angle of elevation of the shot nor

the initial velocity of the ball can be completely and accurately controlled

Errors will occurThe problem we set ourselves was to find the

“most forgiving shot” That is, we seek the shot that will not only

succeed but will be maximally tolerant of error

Our ModelNo air, no friction, no spinThe center of the ball will be confined to the

plane that is perpendicular to the backboard and passes through the center of the basket

Thus, we are disallowing shooting from the side and disregarding azimuthal error

We will call this plane the trajectory planeWe prohibit any contact with the rimWe limit ourselves, initially, to the “swish

shot”Nothing but net!

The Set-UpIn addition to the constraints set out in our

model, we will use parameters designated by letters to denote the dimensions of the court

This will allow us to make general observations not limited to one set of values for these parameters

The Set-Up Here is our list of parameters:R = radius of rimH = height of rim from floord = horizontal distance from point of launch

to front end of rimr = radius of ballh = height of point of launch from floor

The Set-UpWe will endow the trajectory plane (the

previous picture) with a coordinate systemWe will take the origin to be the point of launch Thus, the x-axis will be parallel to the floor but h

units above itThe y-axis will be the vertical line through the

point of launch In this coordinate system, for any choice of

angle and initial velocity, the trajectory of the center of the ball will have the following equation:

22 20

192tan( )

cos ( )y x x

v

Derivation of Trajectory EquationThe previous equation was derived from the

standard parametric representation of a particle launched in a friction-free environment by eliminating the parameter:

2

0 0( ) [ cos( ) ]* [ sin( ) ]*2

gtr t v t i v t j

Important ObservationsRecall

Note that if is held fixed and v0 is increased, the graph of y as a function of x will be elevated in the plane

Similarly, for fixed , decreasing v0 will serve to lower the graph in the plane

This agrees with one’s intuition

22 20

192tan( )

cos ( )y x x

v

The Configuration PlaneThe equation for the trajectory of the center of

the ball depends upon selecting values for and for v0

For each choice, we get a specific trajectory The plane determined by a horizontal axis

indicating the value of and vertical axis indicating the value of v0 will be called the configuration plane.

Each point in the configuration plane will be an ordered pair (, v0 ) which determines a unique trajectory

The Legal RegionEach point in the configuration plane

determines a trajectorySome of these trajectories will yield legal

swish shotsOthers will notWe will call the set of points in the

configuration plane that correspond to proper swish shots, the legal region

Determining the Legal RegionTo determine this region, we shall compute a

greatest lower bound function and a least upper bound function that give, for each value of , the upper and lower bounds on v0

The graphs of these two functions in the configuration plane bound the legal region

The Lower BoundConsider the circle whose center is the

leading edge of the rim and whose radius is the radius of the ball

Any trajectory of the center of the ball that is tangent to the upper right-hand quadrant of this circle will yield a point on the lower bound curve

The Circle About Leading Edge

The Lower BoundWe simultaneously equate the equations of

the circle and the trajectory as well as their derivatives to obtain formulae for the angle of elevation and initial velocity in terms of x

The parametric curve derived in this way yields the lower bound curve in the configuration plane:

( ) ( ( ), ( ))L LL x x V x

The Lower BoundThe components of L(x) from the previous

slide are given by:

2 2

2 2

2 ( )2( )( ) arctan

( )L

r x dH h x dx

x x r x d

2

2 2 2

192( )

( ( ) ( ) tan( ( ))) cos ( ( ))L

L L

xV x

r x d H h x x x

The Upper BoundSimilar to the lower bound function, we

compute the upper bound function that finds, for each choice of angle, , the precise value of v0 that just succeeds in making it below the back edge of the rim

This will coincide with the set of trajectories that are precisely tangent to the lower left-hand quadrant of the circle of radius r about the back edge of the rim

The Upper BoundThis procedure yields a parametric curve in the

configuration plane that gives the upper bound for the value of initial velocity for each choice of angle

We call this function U(x):

( ) ( ( ), ( ))U UU x x V x

The Upper BoundThe components of U(x) from the previous

slide are given by:

2 2

2 2

2 ( 2 )2( ) 2( ) arctan

( 2 )U

r x d RH h x d Rx

x x r x d R

2

2 2 2

192( )

( ( 2 ) ( ) tan( ( ))) cos ( ( ))U

U U

xV x

r x d R H h x x x

The Legal RegionArmed with the upper and lower bound curves, we

can now graph the legal region in the configuration plane

This will necessitate our filling in specific values for the parameter letters we assigned at the beginning

We will choose the standard NBA values: R = 9, r = 4.7, d = 156, H= 120 (all units are inches)

There is no standard value for h, the height of the point of launch, as this will vary from player to player

We shall take h = 72

The Least Forgiving ShotObserve that the upper and lower bound

curves meet at a pointThis point must represent a trajectory that is,

simultaneously, tangent to the circles of radius r about both the front and the back edge of the rim

This shot is the least forgiving shot as any increase or decrease in the value of v0 will result in a failed shot

No error, whatsoever, can be tolerated

We Seek the Most Forgiving ShotIn order to make clear what we mean by the

most forgiving shot, we shall need some definitions

For any point in the legal region of the configuration plane, consider the distances, left and right of that point to the two bounding curves

We shall call the lesser of these two values (and their common value if they are equal) the θ-tolerance of the point

Θ-Tolerance

v0 - ToleranceSimilarly, for any point in the configuration

plane, we find the vertical distances, up and down from the point, to the bounding curves

We call the smaller of the two distances (and their common value if they are equal) the v0 – tolerance of the point

v0 - Tolerance

Inscribed RectangleInscribe a rectangle in the legal region and

draw the vertical line that lies midway between the sides of the rectangle

Now draw the horizontal line that lies midway between the top and bottom

The point where these lines meet is the center of the rectangle

Inscribed Rectangle with Lines

Tolerances within the RectangleObserve that every point on the vertical mid-line has θ-

tolerance that is at least half the width of the rectangle and nearly all points on it have much greater θ-tolerance

Similarly, every point on the horizontal mid-line has v0-tolerance that is at least half the height of the rectangle and nearly all of the points have much greater v0-tolerance

Inscribed Rectangle with Lines

Most Forgiving ShotWe will define the most forgiving swish shot

to be the shot determined by the center of the inscribed rectangle of maximal area

Thus, we are seeking to maximize the product of the smallest θ and v0 - tolerances

Most Forgiving ShotWith the aid of a computer program written

in Maple, I was able to determine that the values for the most forgiving swish shot to be θ = 0.943 radians (about 54 degrees) and v0 = 290.85 inches per second

Most Forgiving Swish Shot

The Bounce ShotSo far, we have concentrated on the swish

shot. But what about the shot that bounces off the backboard and then goes cleanly into the basket without touching the rim?

It turns out that this shot is equivalent to the swish shot that passes through the “phantom basket” that is the reflection of the real basket in the “mirror” of the backboard

The Bounce Shot

The Bounce ShotThe leading edge of the “phantom basket” is

186 inches horizontally from the point of launch

Thus, to find the optimal bounce shot, all we need do is repeat the search for the largest inscribed rectangle in the legal region generated with d = 186 in place of d =156

ConclusionWe conclude with the admission that this

project was an exercise in mathematical analysis. Its purpose was not that of utility

For those who find themselves aggrieved by this admission, I am prepared to accept the referee’s call of “foul” and offer, by way of recompense, one free throw

Thank you