Game Development Reference
In-Depth Information
The Constant Rule
d
dt
k = 0, k is any constant.
The next rule, sometimes known as the sum rule, says that differentia-
tion is a linear operator. The meaning of “linear” is essentially identical to
our definition given in Chapter 5, but let's review it in the context of the
derivative. To say that the derivative is a linear operator means two things.
First, to take the derivative of a sum, we can just take the derivative of
each piece individually, and add the results together. This is intuitive—the
rate of change of a sum is the total rate of change of all the parts added
together. For example, consider a man who moves about on a train. His
position in world space can be described as the sum of the train's position,
plus the man's position in the body space of the train.
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Likewise, his ve-
locity relative to the ground is the sum of the train's velocity relative to
the ground, plus his velocity relative to the train.
Derivative of a Sum
dt
[f(t) + g(t)] =
d
d
dt
f(t) +
d
dt
g(t).
(11.7)
The second property of linearity is that if we multiply a function by some
constant, the derivative of that function gets scaled by that same constant.
One easy way to see that this must be true is to consider unit conversions.
Let's return to our favorite function that yields a hare's displacement as
a function of time, measured in furlongs. Taking the derivative of this
function with respect to time yields a velocity, in furlongs per minute.
If somebody comes along who doesn't like furlongs, we can switch from
furlongs to meters, by scaling the original position function by a factor of
201.168. This must scale the derivative by the same factor, or else the hare
would suddenly change speed just because we switched to another unit.
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Assume that the train tracks are straight, so that the train's body axes are aligned
with the world axes, and no rotation is needed.
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