conic arc function

Question

I would like to have a function $f(x,\alpha)$ looking like this:

• The lower the line, the bigger $\alpha$ is
• The straight line in the middle correspond to $\alpha=0$
• The line forming the upper left corner correspond to $\alpha=-\infty$
• The line forming the lower right corner correspond to $\alpha=\infty$

At the moment, I use the function $f(x) = x^{(e^\alpha)}$, but it's not good. I would like my function to have a symmetry axis corresponding to the dotted line in the picture above. I think it should be a conic

I don't really know how to start to developp the equations. Could you help me find $f(x,\alpha)$ ?

Answer 1

This is the superellipse. For the curves you want, this can be written as

$$ y=(1-(1-x)^{1/p})^p,\quad x\in[0,1] $$

I believe my $p$ is your $\alpha$, but check. My curves look as shown below,