A complete closed-form method for transformation from Cartesian to geodetic coordinates

By introducing the auxiliary variable with respect to the reduced latitude, a new closed-form method for transforming Cartesian to geodetic coordinates has been proposed based on the solution of a special constructed unary quartic equation. The algorithm comes with rigorous and concise procedure of root-finding. Moreover, through theoretical analysis, different approaches with respective pros and cons to determine the geodetic latitude and height have been explored. Besides fast computation, numerical experiments covering the magnitude of the geodetic height from \(- 6.33 \times 10^{6} {\text{m}}\) $-6.33\times {10}^6\text{m}$ to \(10^{10} {\text{m}}\) ${10}^{10}\text{m}$ have also shown that the new method can be operational with high precision at almost any point including the region near or at the pole, the equator and the center of the reference ellipsoid. Considering the accuracy, efficiency and adaptability simultaneously, it is prospective to be applied into computation and inspection on critical occasions in comparison to existing methods.