  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
Gr\"obner 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.
