# Parabolic Coordinates Radius

by bolbteppa
Tags: coordinates, parabolic, radius
 P: 156 Given Cartesian $(x,y,z)$, Spherical $(r,\theta,\phi)$ and parabolic $(\varepsilon , \eta , \phi )$, where $$\varepsilon = r + z = r(1 + \cos(\theta)) \\\eta = r - z = r(1 - \cos( \theta ) ) \\ \phi = \phi$$ why is it obvious, looking at the pictures (Is my picture right or is it backwards/upside-down?) that $x$ and $y$ contain a term of the form $\sqrt{ \varepsilon \eta }$ as the radius in $$x = \sqrt{ \varepsilon \eta } \cos (\phi) \\ y = \sqrt{ \varepsilon \eta } \sin (\phi) \\ z = \frac{\varepsilon \ - \eta}{2}$$ I know that $\varepsilon \eta = r^2 - r^2 \cos^2(\phi) = r^2 \sin^2(\phi) = \rho^2$ ($\rho$ the diagonal in the x-y plane) implies $x = \rho \cos(\phi) = \sqrt{ \varepsilon \eta } \cos (\phi)$ mathematically, but looking at the picture I have no physical or geometrical intuition as to why $\rho = \sqrt{ \varepsilon \eta }$.