I am looking for the way to "quite easily" express particular curves using functional equations.
What's important (supposing the chart's size is 1x1 - actually it doesn't matter in the final result):
- obviously the shape - as shown in the picture;
- there should be exactly three solutions to f(x) = x:
x=0, x close to or equal 0.5 and x=1;
- (0.5,0.5) should be the point of intersection with y = x;
- it would be really nice, if both of the arcs are scalable - as shown in the left example (the lower arc is more significant than the upper one).
I've done some research, but nothing seemed to match my needs; I tried trigonometric and sigmoid functions too, they turned out to be quite close to what I want. I'd be grateful for any hints or even solutions.
P.S. The question was originally asked at stackoverflow and I was suggested to look for help here. Some answers involved splines, cumulative distribution functions or the logistic equation. Is that the way to go?