Is there a quick, geometric, way of writing down (the square root of?) the Cartesian coordinates
$$\begin{align} x^2 &= (a^2+\xi)(a^2+\eta)(a^2+\zeta)/(b^2-a^2)(c^2-a^2)\\ y^2 &= (b^2+\xi)(b^2+\eta)(b^2+\zeta)/(a^2-b^2)(c^2-b^2)\\ z^2 &= (c^2+\xi)(c^2+\eta)(c^2+\zeta)/(a^2-c^2)(b^2-c^2). \end{align}$$
of the ellipsoidal coordinate system by drawing a picture, analogous to the way one derives the Cartesian components of the spherical system by drawing a circle and projecting:
It seems as though there is a trigonometric method for the special case of oblate-spheroidal coordinates
(Also here), so maybe this will help in the quest for ending up with the Cartesian $x^2$ etc... given above.