Symmetric curving function

Question

I'm trying to find an elegant function $y = f(x, a)$, that maps a linear space defined on $[0, 1]$ into a curved one with a curve amount $a$.

$a = -1$ : full knee top left

$a = 0$ : identity

$a = 1$ : full knee bottom right

This function should be symmetric with respect to both diagonals.

Anyone have an idea please ? Many thanks !! :)

Answer 1

Hint: consider $x\longmapsto x^\alpha$, $\alpha\in(0,\infty)$. Define $x\longmapsto f(x,a)$ as a piecewise function using the symmetry condition. Write $a$ as function of $\alpha$.