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Ellipsoidal and Cartesian Coordinates Conversion
|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.
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.