Game Development Reference
In-Depth Information
The curveball shown in Figure 7-24 will break downwards and to the right. The spin on the fast-
ball will tend to loft the flight of the pitch. It will show less break than any of the other types
of pitches.
To evaluate the Magnus force on a fastball according to Equation (7.55), we must deter-
mine the lift coefficient for the baseball. Researchers at the University of California, Davis,
came up with two equations that estimate
C
L
as a function of the rotational spin ratio of the
ball.
6
The equations, shown in Equation (7.58), were created as a best fit to the available exper-
imental data.
r
w
r
v
w
C
=
1.5
for
<
0.1
(7.58a)
L
v
w
=+
r
r
v
w
0.09
0.6
>
0.1
C
for
(7.58b)
L
v
The Drag Coefficient of a Baseball
An estimation of the drag coefficient is necessary to compute the aerodynamic drag experi-
enced by a baseball. Experimental data
6
suggests that
C
D
is approximately 0.5 until a Reynolds
number of between 130,000 and150,000, when the drag coefficient drops sharply due to the
transition of the flow over the baseball from laminar to turbulent. The drag coefficient reaches
a minimum value of about 0.15 at a Reynolds number of about 160,000 and then rises again as
the Reynolds number continues to increase, leveling off at a value of about 0.35 at Reynolds
numbers above 200,000-250,000. The Reynolds number of 160,000 corresponds roughly to a
ball velocity of 32
m/s
(72
mph
).
For game programming purposes, the drag coefficient could be modeled by finding equations
that fit the experimental
C
D
data, including the downward spike that occurs at Re = 160,000.
A simpler, although somewhat less accurate, approach is to assume that
C
D
= 0.5 for
,
C
D
= 0.35 for
Re
<
140 000
,
Re
>
225 000
,
, and decrease
C
D
linearly from 0.5 to 0.35 for
140 000
,
<<
Re
225 000
,
.
Modeling the Hit
The objective of the batter is to hit the ball either over the outfield wall or into the field of play
such that the batter can reach base before he is thrown out. Modeling the collision of the base-
ball bat, shown schematically in Figure 7-25, is a complicated problem. At the point of collision,
both the bat and ball will have three-dimensional velocity components. The ball will certainly
be spinning at the point of impact, and the bat may be rotating as well. The point of impact can
be almost anywhere on the ball depending on the relative positions of the ball and bat at the
time of impact.
Rather than trying to model all of the variables that go into an impact between a baseball
and bat, for game programming purposes it's probably a better idea to simply specify the post-
collision velocity, spin rate, and spin axis of the ball. It's relatively easy to specify velocity and
spin components that would reproduce a single to left field, a pop-up, or any other type of hit
that you would want to model.
