Smooth Interpolation

It is often useful when interpolating to accelerate into the interpolation and/or decelerate out of the interpolation. For example, an object moving from one place to another looks a little better if it accelerates from its stationary state into the move, and then slows down before stopping again. Here I'll talk about how to smooth out a linear interpolation as much (or as little) as you want.

Without any smoothing, you just linearly increase the interpolation factor from zero to one. Instead of this linear movement, let's try increasing the interpolation factor as a function of time f(t). Now let's say that the length in time of the interpolation is 1 (you can easily scale this to any interpolation over any time length), and that the length of the acceleration is a, and the length of the deceleration is b. Then we want the derivative of f(t) to look something like this



Now we just need to determine the height of this trapezoid, such that the area under f'(t) is 1 so that f(1) = 1. The area under the curve is



So for the acceleration (0 ≤ t < a)



And for the linear part (a ≤ t < 1-b)



Our initial condition for this part is that f is continuous at f(a), so



So the linear segment is



And finally, for the deceleration (1-b ≤ t ≤ 1) we have



Again, to satisfy continuity at f(1-b), we need



So the deceleration segment is



And the function with the desired acceleration and deceleration is



Here is a graph of f(t) for various values of a and b.