Editing Geographic coordinate conversion (section)

=== Geodetic to/from ENU coordinates ===
To convert from geodetic coordinates to [[local tangent plane]] ([[Axes conventions#Ground reference frames: ENU and NED|ENU]]) coordinates is a two-stage process:

# Convert geodetic coordinates to ECEF coordinates
# Convert ECEF coordinates to local ENU coordinates

==== From ECEF to ENU ====

To transform from ECEF coordinates to the local coordinates we need a local reference point.  Typically, this might be the location of a radar.  If a radar is located at <math>\left\{X_r,\, Y_r,\, Z_r\right\}</math> and an aircraft at <math>\left\{X_p,\, Y_p,\, Z_p\right\}</math>, then the vector pointing from the radar to the aircraft in the ENU frame is

: <math>
  \begin{bmatrix}x \\ y \\ z\end{bmatrix} =
  \begin{bmatrix}
    -\sin\lambda_r &            \cos\lambda_r           &         0 \\
    -\sin\phi_r\cos\lambda_r & -\sin\phi_r\sin\lambda_r & \cos\phi_r \\
     \cos\phi_r\cos\lambda_r &  \cos\phi_r\sin\lambda_r & \sin\phi_r
  \end{bmatrix}
  \begin{bmatrix}
    X_p - X_r \\
    Y_p - Y_r \\
    Z_p - Z_r
  \end{bmatrix}
</math>

Note: <math>\ \phi</math> is the ''[[geodetic latitude]]''; the ''[[geocentric latitude]]'' is inappropriate for representing [[vertical direction]] for the local tangent plane and must be [[geocentric latitude|converted]] if necessary.

==== From ENU to ECEF ====

This is just the inversion of the ECEF to ENU transformation so

: <math>
  \begin{bmatrix}X_p \\ Y_p \\ Z_p\end{bmatrix} =
  \begin{bmatrix}
    -\sin\lambda_r &  -\sin\phi_r\cos\lambda_r & \cos\phi_r\cos\lambda_r \\
     \cos\lambda_r &  -\sin\phi_r\sin\lambda_r & \cos\phi_r\sin\lambda_r \\
               0 &   \cos\phi_r            & \sin\phi_r
  \end{bmatrix}
  \begin{bmatrix}x \\ y \\ z\end{bmatrix} +
  \begin{bmatrix}X_r \\ Y_r \\ Z_r\end{bmatrix}
</math>