Why Does the Radius in Parabolic Coordinates Involve √(εη)?

[bolbteppa]

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 $\rho = \sqrt{ \varepsilon \eta }$.

 

[fresh_42]

$\varepsilon \eta = (r+z)(r-z)=r^2-z^2=h^2$ and the height should occur at the position $(x,y)$.