If I have an equidissection of a square into various polygons, and I want to map each point on the square to a point on the unit circle such that it each piece (which is not necessarily still a polygon) still has the same area as each other piece.
Generally, I've tried using the square of side length $2$ centered at $0, 0$ and I tried to map $r, \theta$ to $\frac{r}{r_{max}}, \theta$ where $r_{max}$ is the largest $r$ such that the point $r, \theta$ fits inside the square, but respective area is in this case not preserved unless the $\theta$ values of each point are constant, meaning the area is $0$.
Is there some sort of way I can use the Schwarz-Christoffel mapping backwards to map a square to the infinite half-plane and then reapply it to turn that half-plane into a polygon of side-length $\infty$ (i.e. a circle)? And is there any way that would likely preserve respective area?
Is there another mapping I should try?