Transforming Cartesian Coordinates X, Y, Z to Geographical Coordinates φ, λ, h

Abstract

In November 1994, the Intergovernmental Committee on Surveying and Mapping (ICSM) adopted a new geodetic datum for Australia and recommended its progressive implementation nationally. This datum, designated Geocentric Datum of Australia (GDA) - compatible with the Global Positioning System (GPS), a US Department of Defense satellite positioning system - will eventually replace the existing Australian Geodetic Datum (AGD). Consequently, “old” AGD coordinates need to be converted to “new” GDA coordinates, which may require the transformation of Cartesian coordinates X,Y,Z to geographical coordinates φ, λ, h (latitude, longitude, height) related to an ellipsoid of revolution.

Given X,Y,Z, longitude is easily derived, but not so latitude, which requires more sophisticated evaluation - usually by iterative techniques, or complicated direct methods - the height following readily once latitude is obtained. Thus Cartesian-to-geographic transformations revolve around the determination of latitude; this paper reviews published techniques, some quite recent, which may be of use to practitioners.