= Parabolic coordinates
{title2=$x=uv\cos\phi,\quad y=uv\sin\phi,\quad z=(u^2-v^2)/2$}
The displayed three-dimensional parabolic coordinates use $u,v\geq0$ and an azimuthal angle $\phi$. Their coordinate <tangent vectors> have pairwise zero <dot products>, with <scale factors of orthogonal coordinates> $h_u=h_v=\sqrt{u^2+v^2}$ and $h_\phi=uv$. The <Jacobian determinant> is $uv(u^2+v^2)$. Constant $u$ or $v$ gives a family of confocal circular <elliptic paraboloids>; the coordinate system degenerates on the axis where $uv=0$. For example, $0\leq u\leq v\leq1$ fills the cap $(x^2+y^2-1)/2\leq z\leq0$, whose rim is the unit <circle> and whose vertex is $(0,0,-1/2)$.
Back to article page
