An exact conversion from an Earth-centered coordinate system to latitude, longitude and altitude

Present combat systems are being upgraded to properly meet the new capabilities being developed against tactical ballistic missiles. The local coordinate systems are being upgraded to a global (WGS-84) coordinate system so that the correct exchange of threat information with other defending units can be done correctly and in a timely manner. This coordinate system upgrade is also used to properly track these threats, The computation of an Earth-centered coordinate (X_ECC, Y_ECC, Z_ECC) system for a track referenced by altitude, latitude, and longitude is straightforward in biaxial (WGS-84) ellipsoid Earth. The problem arises using the Earth-centered coordinate system for the inverse operation to recompute the altitude, latitude and longitude of the track. This computation can only be done approximately. This article introduces a unique analytic derivation to calculate exactly the latitude, longitude, and altitude of a track from an Earth centered coordinate system. The problem is transformed into solving for a root of an eighth-order polynomial, which can then be reduced and solved by rewriting as a quartic equation. A robust, iterative numerical algorithm for computer implementation is also presented for platforms having computational and memory constraint. An example is given and comparisons made to show the application of these techniques