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:







What i don't understand is why


since cos_pi and sin_pi

What am i missing?

  • (1) This site does not support TeX markup. You can usually get by with code markup (indent lines by four spaces). (2) What is "retaliative distance"?
    – whuber
    Jan 14, 2014 at 22:24
  • @whuber, thanks for your comment, fixed (relative distance).
    – user25788
    Jan 14, 2014 at 23:25
  • For your formulas to be readable, you will need to remove all the TeX markup.
    – whuber
    Jan 15, 2014 at 0:20
  • 1
    @whuber, i hope it is now understandable
    – user25788
    Jan 15, 2014 at 9:22
  • Your formulas appear to be asking about Euler angles but the connection to your question is not evident.
    – whuber
    Jan 15, 2014 at 15:07

1 Answer 1


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.

  • thanks for you answer. I computed the result of R_L and saw that it is correct. My question was refering to R_3(\pi)... why R_3(\pi) is not equal to diag(-1,-1,1), since cos(pi) = -1 and sin(pi) = 0.
    – user25788
    Jan 16, 2014 at 23:29
  • So clearly the R_3(pi) is wrong! Presumably, it's supposed to be R_3(-pi/2). But don't ask me why, because I don't find it particularly useful to decompose R into individual rotations in this case. I prefer just to go straight to R by listing the components of E, N, and U.
    – cffk
    Jan 17, 2014 at 0:52
  • ok, so i will try to compute it and see if i get -pi/2. Strange since i saw pi in more then one book....
    – user25788
    Jan 17, 2014 at 6:42

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