Geodetic coordinates

Definition

Let $\mathcal{E}$ be a reference ellipsoid with semi-axes $a \ge b > 0$ and squared eccentricity $e^2 = 1 - b^2/a^2$. Place it at the origin of a Cartesian coordinate system, with its axis along the $z$-axis. For a geodetic latitude $\varphi \in [-\pi/2, \pi/2]$, set $$N(\varphi) = \frac{a}{\sqrt{1 - e^2 \sin^2 \varphi}}.$$ The point with geodetic coordinates $(\varphi, \lambda, h)$, where $\lambda$ is the longitude and $h \in \R$ the ellipsoidal height, is $$X = (N + h) \cos\varphi \cos\lambda, \quad Y = (N + h) \cos\varphi \sin\lambda, \quad Z = \big((1 - e^2) N + h\big) \sin\varphi,$$ with $N = N(\varphi)$. Geometrically, $h$ is the signed distance from the point to $\mathcal{E}$ along the normal to $\mathcal{E}$. The latitude $\varphi$ is the angle between this normal and the equatorial plane.