= Parabolic polar coordinates
{title2=$x=(u^2-v^2)/2,\quad y=uv$}
= Planar parabolic coordinates
{synonym}
These plane <orthogonal coordinates> use two families of confocal <parabolas>. On the region $y\geq0$, take $u,v\geq0$ and define
$$
u=\sqrt{x+\sqrt{x^2+y^2}},\qquad v=\sqrt{-x+\sqrt{x^2+y^2}}.
$$
Then $x=(u^2-v^2)/2$ 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 $(u^2+v^2)\,du\,dv$, 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.
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