In http://mathproofs.blogspot.com/2005/07/mapping-square-to-circle.html, there is a derivation of the mapping from a unit square to a unit circle. Looking up wikipedia tells me the canonical form of an ellipse is:
$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
and the aforementioned article starts with:
\frac{(x^')^2}{x^2} + \frac{(y^')^2}{b^2} = 1
and proceeds to solve for b.
My question is why is it that $x^2$ is the correct value for $a^2$ in this particular problem?