= Circular torus coordinates
{title2=$\mathbf x=((A+r\sin\alpha)\cos\beta,(A+r\sin\alpha)\sin\beta,r\cos\alpha)$}
For a <ring torus> of major radius $A>0$, <circular torus coordinates> use distance $r$ and angle $\alpha$ in the circular cross-section, and azimuth $\beta$ around the central axis. The coordinate <tangent vectors> are the three partial <derivatives> of the displayed position. They are mutually <orthogonal>, with <scale factors of orthogonal coordinates> $1,r,A+r\sin\alpha$. For $0<r<A$ their ordered <Jacobian determinant> is $r(A+r\sin\alpha)>0$, so $dV=r(A+r\sin\alpha)\,dr\,d\alpha\,d\beta$. At $r=0$ the cross-section angle degenerates. These are tubular coordinates for a circular torus, rather than the confocal coordinate convention also called toroidal coordinates.
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