ECEF From Curvilinear Conversion

The meridian (or the north-south motion) radius of curvature, $$, varies with latitude and is smallest at the equator, where the geocentric radius is the largest, and largest at the poles:

$$

where $$ is the geodetic latitude in radians, $$ is the eccentricity of the reference ellipsoid (WGS 84 - 0.0818191908425), $$ is the equatorial radius (WGS 84 - 6,378,137.0m).

The radius of curvature for east-west motion, $$ is the vertical plane perpendicular to the meridian plane and is not the plane of constant latitude. It varies with latitude and is smallest at the equator:

$$

The Cartesian ECEF position can be obtained from curvilinear position by:

$$

where $$ is the geodetic height or altitude (distance from a body to the ellipsoid surface along the normal to that ellipsoid).

Curvilinear From ECEF Conversion

Conversion from ECEF position to LLA is given by:

$$

where a four-quadrant arctangent function must be used for longitude. Also, since $$ is a function of latitude, the latitude and heigh must be solved iteratively. The approximate closed-form latitude solution (accurate to within 1cm for positions close to the Earth's surface) is given by:

$$

where

$$

$$ of longitude is about 110 km (60 nautical miles) at the equator, and 80 km at $$ latitude.