Game Development Reference
In-Depth Information
often the case where a body of mathematical tools exists to handle con-
tinuous functions (or, equivalently, “signals”), and we wish to apply those
tools to signals with discontinuities. There is a handy mathematical kludge
that can be used to encode a discontinuity in a function such that it can
be integrated. It is known as the Dirac delta, and is usually denoted with
the lowercase letter delta, for example δ(a).
The symbol δ(t) is a special sort of function, which is a spike, or impulse.
Its value is zero everywhere except at t = 0, where it is infinite. But the
actual values of the Dirac delta are not to be taken too literally—what is
more important is that its integral (the “area” of this infinite spike) is equal
to 1. The best way to think about the Dirac delta is as a box centered at
0 with width w and height 1/w. (Other shapes may be chosen, but the
important point is that the shape must have unit area.) The Dirac delta is
defined as the limit of such a function, as the width approaches zero, and
the height approaches infinity, all the while maintaining the unit area. We
like the advice of Bracewell [9], which is to avoid using the term “function”
when referring to special, uh, functions such as the Dirac delta, and instead
use the word “symbol.” This is because whenever we see the Dirac delta
symbol, the entire surrounding expression should be interpreted as a limit.
We are considering the limit as the width approaches zero of some shape
with unit area centered at the origin.
Armed with the Dirac delta, we can differentiate functions with discon-
tinuities. For example, let's say the velocity of a baseball being struck by
a bat at time t = 2 is approximated by the discontinuous function
−130 ft/s t < 2,
130 ft/s
v(t) =
t ≥ 2.
We can differentiate this expression by using the Dirac delta as
v
(t) = 260δ(t − 2),
which can be read as “an impulse of magnitude 260 at 2.” Remember that
the δ symbol is not an ordinary function, and so we interpret the above as
“the limit of a total change in velocity of 260 ft/s that takes place over an
interval of duration ∆t surrounding the time t = 2, as ∆t approaches zero.”
The Dirac delta comes up in a variety of contexts where tools from
continuous math are applied to discontinuous signals. For example, in
graphics, the screen-space image is a signal with inherent discontinuities,
and we need to sample this signal and reconstruct it. The user's input
via the controller is another signal that exhibits discontinuities. The Dirac
delta and other related symbols (such as the ramp function and Heaviside's
step function) are helpful in discussing and manipulating such signals.
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