Coordinate transformation

Hello,

I was wondering if someone can help me with this.

I am trying to transform a circle into a rectangle. Basically, I am looking for the relation between x' and x where x' will represent the points in the new rectangle and x represents the points in the circle.

The problem is related to coordinate transformation. I can understand how to transform a circle into an ellipse by using x'=x/delta and y'=y where delta=R2/R (R2 radii circle and R length of the ellipse semi-axis). However, I am a bit lost in the other transformation. Can someone help? - it is quite important.

Thank you!