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 [itex]\phi[/itex] is the geodetic latitude, [itex]\lambda[/itex] 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?

\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 [itex]\phi[/itex] is the geodetic latitude, [itex]\lambda[/itex] 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?

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