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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.
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.

Figure 1: 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.