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Ellipsoidal and Cartesian Coordinates Conversion

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FundamentalsFundamentals
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.

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.