Approximate formulas

The technical manual for Geocentric Datum of Australia provides direct transformation formulas³ that we reproduce here, after having renamed some parameters to be consistent with our notation and to correct an ambiguity⁴, and after having replaced an equality sign by an approximation sign:

$$\tan{\varphi} \approx \frac{z(1-f) + f(2-f) a_e sin^3u} {(1-f)(r - f(2-f) a_e \cos^3u)}$$ (16)

where

$$\tan u = \frac{z}{r}\left[(1-f) + f(2-f) \frac{a_e}{\sqrt{r^2+z^2}}\right]$$ (17)

Some numerical checks show that these expressions are very good approximation but that they can be applied only for almost spherical bodies. For the Earth, the maximal error are about $10^{-11}$ degrees in latitude and $2\times 10^{-9}$ metres in altitude!

However, when we consider very flat ellipses (for example $f=0.9$), the errors become tremendous (more than $100$ degrees on $\varphi$).