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These plane orthogonal coordinates use two families of confocal parabolas. On the region $y\ge 0$, take $u,v\ge 0$ and defineThen $x=(u^2-v^2)/2$ and $y=uv$. The coordinate tangent vectors $(u,v)$ and $(-v,u)$ have zero dot product, and both have length $\sqrt{u^2+v^2}$. Thus the Jacobian determinant is $u^2+v^2$, the area element is $(u^2+v^2)dudv$, and these orthogonal coordinates degenerate only at the origin. Fixed $u=a>0$ gives $x=(a^2-y^2/a^2)/2$; fixed $v=b>0$ gives $x=(y^2/b^2-b^2)/2$. This normalization is a rescaling of the plane section of parabolic cylindrical coordinates.

Without a restricted parameter domain, simultaneous sign reversal $(u,v)\mapsto (-u,-v)$ preserves the image, so these planar parabolic coordinates are globally two-to-one. The inverse function theorem supplies local inverse branches away from the origin. The nonnegative quadrant is one useful branch chart for the upper half-plane.

The partial derivatives of the parametrization areTheir cross product isTaking its euclidean norm gives the surface area elementThe two poles and duplicated seam have zero surface area, so integrating over the stated parameter domain counts the surface area once. With the change of variables $u=\cos \theta$ and the supplied definite integral,Here $-\ln (\sqrt{2}-1)=\ln (\sqrt{2}+1)$ because $(\sqrt{2}-1)(\sqrt{2}+1)=1$.