Elastic Stretch Data in 2D

Hi everyone,

Say, I have a data as shown in the figure below.

I want to strecth this data to a perfect square.

I got the fitting curve for the arc parts, which is:

    xx1 = 0:.001:5-.001;
    xx2 = flip(-xx1);
    xx = [xx2 xx1];  
    p1 =       0.068;
    p2 =    6.28e-17;
    p3 =         3.3;
    fit = (p1*xx.^2 + p2*xx + p3)';
    plot(xx,fit,'r');

Adding this curve to the original data wouldn't work. I want to stretch it elastically. It's kind of difficult to explain. I hope I could explain it.