Game Development Reference
In-Depth Information
We can create another common and useful distribution in a similar fashion to the
linear distribution above. By inserting an exponent into the formula, thus making
the formula a quadratic function such as we discussed in Chapter 10, we create one
of the veritably plethoric varieties of
parabolic distributions
. The linear distribu-
tions have a subtle advantage of expression over the uniform distributions in that
the probability value changes over the range. Similarly, a parabolic distribution is
slightly more expressive than a linear distribution in that the
rate
of change in
probability changes over the range (Figure 11.18).
FIGURE 11.18
Despite starting and ending at the same pair of locations
as the linear function
y
=
x
(dashed line), the parabolic probability distribution
y
=
x
2
/100 shows significantly different characteristics.
We can manipulate parabolic distributions in the ways that we covered in the
previous chapter. First, by changing the exponent, we can dramatically change the
inflection of the curve. For example, raising
x
to the power of three rather than two
causes a more pronounced “corner� to appear. We can also utilize non-integer ex-
ponents (e.g.,
x
1.7
) to fine-tune the shape of the curve.
By using exponents less than one, we can reverse the inflection of the curve. For
example, the formula x
.5
10 (Figure 11.19) starts by climbing quickly and tapering
off. This curve is reminiscent of some of the decreasing marginal utility curves that
we discussed in Chapter 8.
×
