Suppose I have two curves $y=x^2$, $y=2x^2$ in one $x,y$ coordinate system and limit $0\leq x\leq 1$. What I want to do is transforming this two curves to a new $u,v$ coordinate system. In this new coordinate system. these two curves are straight line. How could I find the transformation component? $(u=f(x,y), v=g(x,y))$
For parallel curve like $y=x^2$, $y=x^2+1$, it's easy to transform to two parallel straight line in the new coordinate system. I just set $u=x,v=y-x^2$ to achieve that. However, it's hard for me to find the transformation component for non-parallel curve.