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# Ellipsoidal and Cartesian Coordinates Conversion

Fundamentals | |
---|---|

Title | Ellipsoidal and Cartesian Coordinates Conversion |

Author(s) | J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain. |

Level | Advanced |

Year of Publication | 2011 |

The ECEF cartesian coordinates can be expressed in the ellipsoidal coordinates , where and are, respectively, the latitude and longitude from the ellipsoid, and the height above it. Figure 1 illustrates the relation between Cartesian and ellipsoidal coordinates.

## From Ellipsoidal to Cartesian coordinates

The Cartesian coordinates of a point can be obtained from the ellipsoidal coordinates by the next expressions:

where is the radius of curvature in the prime vertical:

and where the eccentricity is related with the semi-major axis , the semi-minor axis and the flattening factor by:

## From Cartesian to Ellipsoidal coordinates

The ellipsoidal coordinates of a point can be obtained from the cartesian coordinates as follows:

The longitude is given by:

The latitude is computed by an iterative procedure.

**1.**The initial value is given by:

- with .

**2.**Improved values of , as well as the height , are computed iterating in the equations:

- The iterations are repeated until the change between two successive values of are smaller than the precision required.