Constructing Lorenz-like curves In economics, the Lorenz curve measures economic inequality in countries. The probability density function given by wikipedia is: ![enter image description here]( The further the Lore...--prophetes.ai

Constructing Lorenz-like curves In economics, the Lorenz curve measures economic inequality in countries. The probability density function given by wikipedia is: ![enter image description here]( The further the Lorenz curve is from the line of equality the more inequality there exists. For a project I'm looking to construct a function family or parametric function family ![enter image description here]( that resembles the Lorenz distribution, in the unit square, $ \Bbb R^2 (0,1) \times(0,1), $ with a variable and a smoothly varying parameter that is symmetric about the line $y=x$ and $ y=1-x $, such that the function smoothly transitions from itself to its inverse as the parameter is varied and as it crosses the line $y=x$. The family $ f_n(x)= x^n $ almost works but it is not symmetric about the line $y=1-x$. Also I'm not sure if the Lorenz curve admits an inverse but if it did that would help me a lot and would probably go a long way in answering my question. Thanks.

This system of symmetric superellipse equations works well:

$ x^s+y^s=1 $

$ (1-x)^s+(1-y)^s=1 $

$s \in \Bbb R (1,\infty). $