geod2geoc

Convert geodetic latitude to geocentric latitude

Syntax

gc = geod2geoc(gd, h)
gc = geod2geoc(gd, h, model)
gc = geod2geoc(gd, h, f, Re)


Description

gc = geod2geoc(gd, h) converts an array of m geodetic latitudes, gd, and an array of mean sea level altitudes, h, into an array of m geocentric latitudes, gc. h is in meters. Latitude values can be any value. However, values of +90 and -90 may return unexpected values because of singularity at the poles.

gc = geod2geoc(gd, h, model) is an alternate method for converting from geodetic to geocentric latitude for a specific ellipsoid planet. Currently only 'WGS84' is supported for model. Latitude values can be any value. However, values of +90 and -90 may return unexpected values because of singularity at the poles.

gc = geod2geoc(gd, h, f, Re) is another alternate method for converting from geodetic to geocentric latitude for a custom ellipsoid planet defined by flattening, f, and the equatorial radius, Re, in meters. Latitude values can be any value. However, values of +90 and -90 may return unexpected values because of singularity at the poles.

Examples

Determine geocentric latitude given a geodetic latitude and altitude:

gc = geod2geoc(45, 1000)


gc =

   44.8076

Determine geocentric latitude at multiple geodetic latitudes and altitudes, specifying WGS84 ellipsoid model:

gc = geod2geoc([0 45 90], [1000 0 2000], 'WGS84')


gc =

         0
   44.8076
   90.0000

Determine geocentric latitude at multiple geodetic latitudes, given an altitude and specifying custom ellipsoid model:

f = 1/196.877360;
Re = 3397000;
gc = geod2geoc([0 45 90], 2000, f, Re)


gc =

         0
   44.7084
   90.0000

Assumptions and Limitations

This implementation generates a geocentric latitude that lies between ±90 degrees.

Algorithms

The geod2geoc function converts a geodetic latitude (μ) into geocentric latitude (λ), where:

• λ — Geocentric latitude

• μ — Geodetic latitude

• h — Height from the surface of the planet

• f — Flattening

• a — Equatorial radius of the plant (semi-major axis) (Re)

Given the geodetic latitude (μ) and the height from the surface of the planet (h), this block first calculates the geometric properties of the planet.

$$\begin{array}{l}{e}^2=\frac{f}{(2-f)}\\ N=\frac{a}{\sqrt{1-e^2\sin {(\mu )}^2})}.\end{array}$$

It then calculates the geocentric latitude from the point's distance from the polar axis (ρ) and distance from the equatorial axis (z).

$$\begin{array}{l}\rho =(N+h)\sin (\mu )\\ z=(N(1-e^2)+h)\sin (\mu )\\ \lambda ={\tan }^{-1}\left(\frac{z}{\rho }\right).\end{array}$$

References

Stevens, B. L., and F. L. Lewis, Aircraft Control and Simulation, John Wiley & Sons, New York, NY, 1992