Look for a function that gives a symmetrical curve with flexible curvature

Question

I am searching for a function that would relate two variables approximately like the graphs below, but with a smooth curve.

The joint of the dark line above is where users can click with a mouse, thus specifying the function to be applied.

Both my variables, Tamanho (size) and Nº spp (species number) are within the range [0,1].

The curve should become an almost square near the corners, i.e. (Nº spp ~ 0 and Tamanho ~ 1) or (Nº spp ~ 1 and Tamanho ~ 0).

I have tried exponential functions, but they are not symmetrical.

I have also tried quadratic curves, but they goes outside the range [0,1] for Tamanho, depending on where you click.

Question:

Does such function I'm looking for even exist? If not, what options do I have?

I'm going to include them all in my program (broken line, exponential...), so the bigger the number of options, the better.

Answer 1

Here is a perfect curve that meets all the requirements:

$$(1+a^2)(y-x)^2=a^2+(x+y)^2$$

where $a$ is a parameter that adjusts how sharp the turn is. The sample plot below is for $a=0.05$.

As seen in the graph, the function exists. It has the desired symmetry and stays within the unit range. In the limit $a\rightarrow 0$, a 90-degree corner turn (nearly a square) can be produced.

$$a=0.05$$

Answer 2

It can’t exist as a function $y=f(x)$ if you impose the graph to be a square if you click of the bottom right corner. Indeed for the graph you draw, there is an infinite number of images of $x =1$.