This paper presents two new direct symbolic-numerical algorithms for the transformation of Cartesian coordinates into geodetic coordinates considering the general case of a triaxial reference ellipsoid. The problem in both algorithms is reduced to finding a real positive root of a sixth degree polynomial. The first approach consists of algebraic manipulations of the equations describing the geometry of the problem and the second one uses Grobner bases. In order to perform numerical tests and accurately compare efficiency and reliability, our algorithms together with the iterative methods presented by M. Ligas (2012) and J. Feltens (2009) have been implemented in C++. The numerical tests have been accomplished by considering 10 celestial bodies, referenced in the available literature. The obtained results clearly show that our algorithms improve the aforementioned iterative methods, in terms of both efficiency and accuracy.

中文翻译：

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在三轴椭球上从笛卡尔坐标直接转换为大地坐标

考虑到三轴参考椭球的一般情况，本文提出了两种新的直接符号数字算法，用于将笛卡尔坐标转换为大地坐标。两种算法中的问题都简化为找到六次多项式的实数正根。第一种方法包括对描述问题几何的方程进行代数运算，第二种方法使用 Grobner 基。为了进行数值测试并准确比较效率和可靠性，我们的算法以及 M. Ligas (2012) 和 J. Feltens (2009) 提出的迭代方法已在 C++ 中实现。数值测试是通过考虑现有文献中引用的 10 个天体来完成的。