[GIS] Trasformation from ECEF to ENU

coordinate systemcoordinates

I am trying to transform a local vector rECEF from ECEF coordinate system to ENU coordinate system (rENU) where

ECEF = earth-centered earth-fixed coordinate system

ENU = East, North, Up coordinates, a local Earth based coordinate system

I saw in more then one book the following equation:

r_enu

where

R_L,

r_3_theta

and

R_2_theta

What i don't understand is why

R_3_pi

since cos_pi and sin_pi

What am i missing?

Best Answer

The easiest way to check that this is right is to multiply out the matrices to get

     [     -sin(lambda)            cos(lambda)          0     ]
R =  [ -sin(phi)*cos(lambda)  -sin(phi)*sin(lambda)  cos(phi) ]
     [  cos(phi)*cos(lambda)   cos(phi)*sin(lambda)  sin(phi) ]

The three rows are the unit vectors E, N, and U in ECEF coordinates; so when you form R.r_ecef you are computing the components of r_enu by resolving r_ecef into its ENU coordinates.

Incidently, you are missing the translation component in this conversion. The origin of the ENU system is normally on the surface of the ellipsoid while the origin of ECEF is the center of the earth.

See, for example, Other Earth-based coordinate systems and Conversion calculations at Geodetic datum.

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