Parabolic polar coordinates ($x=(u^2-v^2)/2,y=uv$)

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.

For planar parabolic coordinates and $g(u,v)=f(x(u,v),y(u,v))$, the multivariable chain rule gives $g_u=u{f}_x+v{f}_y$ and $g_v=-v{f}_x+u{f}_y$. Differentiating once more and adding cancels the first partial derivatives and the mixed partial derivatives:Consequently, on every local coordinate patch away from the origin, $f$ is a harmonic function if and only if $g$ is a harmonic function. The Laplacian changes by a positive scale factor, so solutions of the Laplace equation are preserved. The restriction away from the origin is needed to divide by the Jacobian determinant.