Fast transform from geocentric to geodetic coordinates T. Fukushima National Astronomical Observatory, 2-21-1, Ohsawa, Mitaka, Tokyo 181-8588, Japan e-mail: toshio.fukushima@nao.ac.jp; Tel.: +81 422 41 3613; Fax: +81 422 41 3793 Received: 13 November 1998 / Accepted: 27 August 1999 Abstract. A new iterative procedure to transform geo- centric rectangular coordinates to geodetic coordinates is derived. The procedure solves a modi®cation of Borkowski's quartic equation by the Newton method from a set of stable starters. The new method runs a little faster than the single application of Bowring's formula, which has been known as the most ecient procedure. The new method is suciently precise because the resulting relative error is less than 10 15 , and this method is stable in the sense that the iteration converges for all coordinates including the near-geocenter region where Bowring's iterative method diverges and the near-polar axis region where Borkowski's non- iterative method suers a loss of precision. Key words. Coordinate transformation Geocentric coordinates Geodetic coordinates Newton method 1 Introduction The conversion from geocentric rectangular to geodetic coordinates is a basic but nontrivial problem encoun- tered frequently in geodesy and positional astronomy. The existing methods to do it are classi®ed into three categories; (1) the approximate series expansion (e.g. Ta 1985); (2) the numerical iterative procedure (e.g. Heiskanen and Moritz 1967); and (3) the closed analytical formula (e.g. Vanicek and Krakiwski 1982). In his review (Borkowski 1989), Borkowski presented two new methods, one iterative and one non-iterative, and claimed that they are superior to the existing methods in accuracy and/or simplicity. Unfortunately, this work lacks a comparison with the well-known Bowring's formula (Bowring 1976), which has been known as the most ecient method. In fact, Laskowski (1991) shows that Bowring's method, which he de®ned as the iteration of Bowring's formula twice at most, is 30% faster than the iterative method of Borkowski. We have recently found a systematic way to construct stable and fast starters for the Newton method and applied it successfully in developing fast procedures for solving Kepler's equations (Fukushima 1997a, b, 1998). The last work, in particular, includes the accelerated solution of Barker's equation, which is a sort of cubic equation. As a natural extension, we applied the same approach to a modi®cation of the quartic equation Borkowski adopted, and developed a new method to solve it. The new method is: (1) fast in the sense it runs a little faster than the single application of Bowring's formula (see Table 1); (2) precise because it achieves relative errors of order of 10 15 in the double-precision calculation, which means 10 nm on the Earth's surface; and (3) stable since it converges with any possible combination of input coordinates and the parameters of the reference ellipsoid. It is noteworthy that the CPU time of the new method, which requires 3±4 iterations in average, is al- most the same as that of Bowring's non-iterative meth- od. This mainly comes from the fact that each iteration in the new method needs a small number of arithmetic operations and requires no call for transcendental functions. 2 Method The core of the transformation from the geocentric rectangular coordinates x; y ; z to the geodetic coordi- nates u; k; h is the conversion from p; z to u; h, where p x 2 y 2 p 1 and p and z are restricted to be positive. The conversion consists of: (1) the solution of the latitude equation p sin u z cos u e 2 N sin u cos u 2 with respect to u or its equivalent; and (2) the computation of h from thus solved u or others. Here, Journal of Geodesy (1999) 73: 603±610