International Global Navigation Satellite Systems Association
IGNSS Symposium 2018
Colombo Theatres, Kensington Campus, UNSW Australia
7 – 9 February 2018
Approximation solutions to the Cartesian to geodetic coordinate
transformation problem
Hatem Hmam
Defence Science and Technology Group, Australia
hatem.hmam@dsto.defence.gov.au
ABSTRACT
This paper presents several approximation algorithms, which convert
Cartesian coordinates to geodetic coordinates. All the proposed
methods are based on the regular perturbation expansion of the
reduced latitude tangent. After substitution in the latitude equation, it
is shown that the developed solutions can readily be expressed as a
bilinear form on the vector space,
, where is the perturbation
expansion order. The 5
th
order perturbation expansion is later used to
generate a more computationally efficient algorithm and yet achieves
sub-millimeter-level coordinate conversion accuracy. Finally the
conversion accuracy and computational efficiency of all proposed
methods are compared with those of popular iterative algorithms.
KEYWORDS:
Geodetic coordinates, Perturbation, latitude equation,
WGS84, Geodetic ellipsoid
1. INTRODUCTION
The ubiquitous use of satellite based positioning receivers embedded in smart phones,
vehicles, drones and many other devices, has emphasized the need to implement numerically
stable and efficient algorithms for the transformation between the Cartesian rectangular
coordinates and the geodetic coordinates (. This requirement becomes critical
in applications where such a transformation is executed at high frequency, particularly when
only limited processing resources are available on-board the device or machine.
The computation of Cartesian coordinates given their geodetic counterparts is
straightforward, but the reverse transformation requires some computational effort. Similar to
many research approaches in this area, focus is placed on solving for the latitude, , and
altitude, , in the meridian plane (ie. determination of ( given
and ).
Computing the longitude, , is straightforward and non-iterative (see Ligas et al (2011)).
The literature on the Cartesian to geodetic transformation is extensive and ranges from exact
methods to iterative or approximation based algorithms (see for example Gerdan (1999) for a
