Game Development Reference
In-Depth Information
shift is so dramatic that the actual boundaries are based on the decimal precision
that we use for
x.
If we are not using very precise values of
x
, those extreme values
of
y
do not come into play. For example, using
e
as the root (as shown in Figure
10.8),
y
< -5 only when
x
< 0.0057. Therefore, if we only concerned ourselves with
two decimal places, we would never see values of
y
less than -5 or greater than 5.
Because of this, one common shift we can perform is to move the curve up by
five points. This means the effective range of the logit function becomes 0 to 10
rather than -5 to 5. For completeness, the formula would be
By no means are the above functions an exhaustive list. What they do provide is a
number of core building blocks from which to work. By using the basics, tweaking
them with coefficients, and even combining multiple types into the same equation,
we can create many very distinct curves.
As we will see over the next few chapters, fine-tuning a function to fit a need is a
skill that is core to constructing proper behavioral mathematics. Often, constructing
the right curve takes a lot of trial and error. I very much recommend using a tool
such as Excel to construct formulas and examine the resulting graphs.
When all else fails, there are ways of handcrafting data points that break com-
pletely out of the formula-based approach. We shall cover those in Chapter 12.
