math – easing functions for bell curves?
There are lots of common bell-shaped functions f on [0, 1]; I assume you want them to satisfy f(0) = f(1) = f(0) = f(1) = 0 and f(1/2) = 1. Examples:
Any symmetric beta distribution density function, for any parameters α = β > 1, is bell-shaped and has zero derivative at the endpoints. That is,
f(x) = 4^α * x^(α - 1) * (1 - x)^(α - 1), where
4^αis a constant to scale it so that it goes up to 1:
Pick a segment of a sinusoidal function, starting and ending at adjacent troughs, and translating/scaling as desired. Example:
f(x) = (sin(2 * π * (x - 1/4)) + 1) / 2: