I'm looking for a curve that has a slow start and a slow end, as an audio workstation put it. It fails to mention, however what kind of curve it is.
More accurately, I'm looking for a curve like this:
- Begins at $x = a$, ends at $x = a+t$, so has length $t$
- $f'(a) = 0$, $f'(a+t) = 0$
- Bonus if $f'(x)=0$ for $x \in ]-\infty,a]$ and $x \in ]a+t,\infty]$ so one can avoid partially defining
- Bonus if the steepness of the middle point can be altered
The curve looks something like this:
A half of a sine wave is pretty close, but not quite. I'm sure there is even a name for this kind of thing, but I've never encountered it before.
Look into the logistic function $$f(x)=\frac{1}{1+e^{-ax}}$$ for restricted domain (if you want a start and end point). The steepness of the curve at the PoI can be altered by adjusting $a$. This doesn’t satisfy exactly $f’(a)=0$ but it’s a decent approximation
https://en.m.wikipedia.org/wiki/Logistic_function