3.2 Local ellipsoidal radius

The local ellipsoidal radius $r$ is computed as:

$$r(\varphi) = \sqrt{x^{2}+y^{2}+z^{2}} = a\sqrt{1-\frac{e^{2}(1-e^{2})\sin^{2}\varphi}{1-e^{2}\sin^{2}\varphi}}.$$ (12)

An approximation to this radius $r$ is given by (Heiskanen and Moritz, 1967):

$$r(\varphi) = a(1 - f \sin^{2}\varphi).$$ (13)