conic arc function I would like to have a function $f(x,\alpha)$ looking like this: ![each line correspond to a different value of $\\alpha$]( * The lower the line, the bigger $\alpha$ is * The straight line in ...--prophetes.ai

conic arc function I would like to have a function $f(x,\alpha)$ looking like this: ![each line correspond to a different value of $\\alpha$]( * 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 $f(x,\alpha)$ ?

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,

![Superellipse variation](