Direction Cosine Matrix ECEF to NED

Convert geodetic latitude and longitude to direction cosine matrix


Utilities/Axes Transformations


The Direction Cosine Matrix ECEF to NED block converts geodetic latitude and longitude into a 3-by-3 direction cosine matrix (DCM). The DCM matrix performs the coordinate transformation of a vector in Earth-centered Earth-fixed (ECEF) axes (ox0, oy0, oz0) into a vector in north-east-down (NED) axes (ox2, oy2, oz2). The order of the axis rotations required to bring this about is:

  1. A rotation about oz0 through the longitude (ι) to axes (ox1, oy1, oz1)

  2. A rotation about oy1 through the geodetic latitude (μ) to axes (ox2, oy2, oz2)


Combining the two axis transformation matrices defines the following DCM.


Inputs and Outputs

InputDimension TypeDescription


2-by-1 vectorContains the geodetic latitude and longitude, in degrees. Latitude and longitude values can be any value. However, latitude values of +90 and -90 may return unexpected values because of singularity at the poles.
OutputDimension TypeDescription


3-by-3 direction cosine matrixTransforms ECEF vectors to NED vectors.


The implementation of the ECEF coordinate system assumes that the origin is at the center of the planet, the x-axis intersects the Greenwich meridian and the equator, the z-axis is the mean spin axis of the planet, positive to the north, and the y-axis completes the right-hand system.


Stevens, B. L., and F. L. Lewis, Aircraft Control and Simulation, John Wiley & Sons, New York, 1992.

Zipfel, P. H., Modeling and Simulation of Aerospace Vehicle Dynamics, AIAA Education Series, Reston, Virginia, 2000.

“Atmospheric and Space Flight Vehicle Coordinate Systems,” ANSI/AIAA R-004-1992.

Introduced before R2006a