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 \(- 6.33 \times 10^{6} {\text{m}}\) to \(10^{10} {\text{m}}\) 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.

通过引入关于缩减纬度的辅助变量，基于特殊构造的一元四次方程的解，提出了一种将笛卡尔坐标变换为大地坐标的新闭式方法。该算法具有严谨、简洁的求根过程。此外，通过理论分析，探讨了确定大地纬度和高度的不同方法，各有利弊。除了快速计算之外，数值实验涵盖了从\(- 6.33 \times 10^{6} {\text{m}}\)到\(10^{10} {\text{m}}\ ）还表明，新方法几乎可以在任何点上高精度运行，包括极点附近或极点、赤道和参考椭球中心的区域。同时考虑精度、效率和适应性，与现有方法相比，有望应用于关键场合的计算和检测。