Is There a Paradox in the Equation for Oblate Spheroids?

[ianyappy]

I'm a little confused by what appears to me to be a paradox. I understand that the equation for an oblate spheroid is given by
\begin{equation}
\frac{x ^ 2+y^2}{a^2} + \frac{z ^ 2}{c^2}= 1, c < a
\end{equation}
where a and c are the semi-major and semi-minor axes respectively.
However, the definitions for the WGS 84 oblate spheroid model of the Earth using the ECEF <--> geodetic latitude/longitude/height are (Wiki Link) the following equations
\begin{equation}
x = N(\phi) cos\phi cos\lambda \\
y = N(\phi) cos\phi sin\lambda \\
z = N(\phi)(1-e^2) sin \phi \\
N(\phi) = \frac{a}{\sqrt{1-e^2 sin^2\phi}}
\end{equation}
where \phi is the geodetic latitude, \lambda is the longitude, e is the eccentricity and a is the semi-major axis. I removed the height parameter as I am only concerned with the spheroid surface.

I have also read a paper which states the equation relating the ECEF X,Y,Z is
\begin{equation}
\frac{x ^ 2+y^2}{N(\phi)^2} + \frac{z ^ 2}{[N(\phi)(1-e^2)]^2}= 1
\end{equation}

The last equation has denominators that depends on the geodetic latitude while the first equation for the oblate spheroid has denominators which are constant. Is there something wrong?

 

[haruspex]

The equation you quote for an oblate spheroid uses geocentric latitude. I believe the equations on the Wiki page use geodetic latitude. There is a comment about that on the page you linked.

 

[ianyappy]

Well yes, but isn't ECEF independent of geodetic/geocentric coordinates? Then in the last equation, this seems to suggest that x,y,z must fit a different equation for different geodetic latitudes.