Direct transformation from geocentric cartesian coordinates to geodetic latitude and ellipsoidal height

In analysing the geometric relation between the geocentric coordinates and the geodetic latitude and ellipsoidal height, a tangent (or cotangent) quadratic equation of geodetic latitude is created, following which the geodetic latitude can be solved directly. We have found in the calculation of latitude from the tangent equation that the larger the absolute value of latitude is, the higher the accuracy will be, but the calculation from the cotangent equation is just the opposite. Similar phenomenon also occurs in the calculation of ellipsoidal height. Therefore, we take corresponding tactics to ensure the high accuracy and stability in the calculations of geodetic latitude and ellipsoidal height. With the method in the paper, the geodetic latitude and ellipsoidal height of any point in space from the Earth's centre to the height of Earth synchronous orbit satellite can be uniquely determined. The maximum errors for the calculations of geodetic latitude and ellipsoidal height are smaller than 4 x 10 (-12) deg and 4 x 10 (-8) m, respectively.