Parabolic cylindrical coordinates ($x=u^2-v^2,y=2uv,z=w$)

These orthogonal coordinates extend two confocal families of parabolas parallel to an axis. In the displayed scaling their scale factors of orthogonal coordinates are $h_u=h_v=2\sqrt{u^2+v^2}$ and $h_w=1$, so the Laplacian is $\Delta \varphi =({\varphi }_{uu}+{\varphi }_{vv})/[4(u^2+v^2)]+{\varphi }_{ww}$ away from $u=v=0$. With $u\ge 0$, signed $v$ distinguishes the upper and lower half-planes; the two signs at $u=0$ describe the same negative $x$ half-axis. Fixed positive $u^2$ or $v^2$ describes a parabolic cylinder. This differs from the rotational three-dimensional parabolic coordinates, whose coordinate surfaces are elliptic paraboloids.