Geodetic coordinates

Geodetic vs. geocentric coordinates

Geodetic latitude and geocentric latitude have different definitions. Geodetic latitude is defined as the angle between the equatorial plane and the surface normal at a point on the ellipsoid, whereas geocentric latitude is defined as the angle between the equatorial plane and a radial line connecting the centre of the ellipsoid to a point on the surface (see figure). When used without qualification, the term latitude refers to geodetic latitude. For example, the latitude used in geographic coordinates is geodetic latitude. The standard notation for geodetic latitude is φ. There is no standard notation for geocentric latitude; examples include θ, ψ, φ′.

Similarly, geodetic altitude is defined as the height above the ellipsoid surface, normal to the ellipsoid; whereas geocentric altitude is defined as the distance to the reference ellipsoid along a radial line to the geocenter. When used without qualification, as in aviation, the term altitude refers to geodetic altitude (possibly with further refinements, such as in orthometric heights). Geocentric altitude is typically used in orbital mechanics (see orbital altitude).

If the impact of Earth's equatorial bulge is not significant for a given application (e.g., interplanetary spaceflight), the Earth ellipsoid may be simplified as a spherical Earth, in which case the geocentric and geodetic latitudes are equal and the latitude-dependent geocentric radius simplifies to a global mean Earth's radius (see also: spherical coordinate system).

Conversion

Given geodetic coordinates, one can compute the geocentric Cartesian coordinates of the point as follows:

${\displaystyle \begin{array}{cc}X& =(N+h)\cos \varphi \cos \lambda \\ Y& =(N+h)\cos \varphi \sin \lambda \\ Z& =\left(\frac{b^2}{a^2}N+h\right)\sin \varphi \end{array}}$

where a and b are the equatorial radius (semi-major axis) and the polar radius (semi-minor axis), respectively. N is the prime vertical radius of curvature, function of latitude ϕ:

${\displaystyle N=\frac{a^2}{\sqrt{a^2{\cos }^2\varphi +b^2{\sin }^2\varphi }},}$

In contrast, extracting ϕ, λ and h from the rectangular coordinates usually requires iteration as ϕ and h are mutually involved through N:

${\displaystyle \lambda =\mathrm{atan2}(Y,X)}$.

${\displaystyle h=\frac{p}{\cos \varphi }-N,}$

${\displaystyle \varphi =\arctan \left((Z/p)/(1-e^{2}N/(N+h))\right).}$

where ${\displaystyle p=\sqrt{X^2+Y^2}}$. More sophisticated methods are available.