Geographic coordinate conversion
In geodesy, conversion among different geographic coordinate systems is made necessary by the different geographic coordinate systems in use across the world and over time.
[1] A geographic coordinate transformation is a translation among different geodetic datums.
Both geographic coordinate conversion and transformation will be considered in this article.
are just temporary variables to handle both positive and negative values properly.
is eliminated by subtracting and The following holds furthermore, derived from dividing above equations: The orthogonality of the coordinates is confirmed via differentiation: where (see also "Meridian arc on the ellipsoid").
The conversion of ECEF coordinates to longitude is: where atan2 is the quadrant-resolving arc-tangent function.
The conversion for the latitude and height involves a circular relationship involving N, which is a function of latitude: It can be solved iteratively,[4][5] for example, starting with a first guess h≈0 then updating N. More elaborate methods are shown below.
[6][7] The following Bowring's irrational geodetic-latitude equation,[8] derived simply from the above properties, is efficient to be solved by Newton–Raphson iteration method:[9][10] where
Bowring showed that the single iteration produces a sufficiently accurate solution.
, derived from the above, can be solved by Ferrari's solution[11][12] to yield: A number of techniques and algorithms are available but the most accurate, according to Zhu,[13] is the following procedure established by Heikkinen,[14] as cited by Zhu.
, then the vector pointing from the radar to the aircraft in the ENU frame is Note:
References such as the DMA Technical Manual 8358.1[15] and the USGS paper Map Projections: A Working Manual[16] contain formulas for conversion of map projections.
There are transformations that directly convert geodetic coordinates from one datum to another.
There are also grid-based transformations that directly transform from one (datum, map projection) pair to another (datum, map projection) pair.
[19] A fourteen-parameter Helmert transform, with linear time dependence for each parameter,[19]: 131-133 can be used to capture the time evolution of geographic coordinates dues to geomorphic processes, such as continental drift[20] and earthquakes.
[21] This has been incorporated into software, such as the Horizontal Time Dependent Positioning (HTDP) tool from the U.S.
[22] To eliminate the coupling between the rotations and translations of the Helmert transform, three additional parameters can be introduced to give a new XYZ center of rotation closer to coordinates being transformed.
[19]: 134 The Molodensky transformation converts directly between geodetic coordinate systems of different datums without the intermediate step of converting to geocentric coordinates (ECEF).
[24] It requires the three shifts between the datum centers and the differences between the reference ellipsoid semi-major axes and flattening parameters.
Grid-based transformations directly convert map coordinates from one (map-projection, geodetic datum) pair to map coordinates of another (map-projection, geodetic datum) pair.
HARNs are also known as NAD 83/91 and High Precision Grid Networks (HPGN).
[27] Subsequently, Australia and New Zealand adopted the NTv2 format to create grid-based methods for transforming among their own local datums.
The NOAA provides a software tool (as part of the NGS Geodetic Toolkit) for performing NADCON transformations.
[28][29] Datum transformations through the use of empirical multiple regression methods were created to achieve higher accuracy results over small geographic regions than the standard Molodensky transformations.
[30] The standard NIMA TM 8350.2, Appendix D,[31] lists MRE transforms from several local datums to WGS 84, with accuracies of about 2 meters.
[32] The MREs are a direct transformation of geodetic coordinates with no intermediate ECEF step.
are modeled as polynomials of up to the ninth degree in the geodetic coordinates
could be parameterized as (with only up to quadratic terms shown)[30]: 9 where with similar equations for
coordinate pairs for landmarks in both datums for good statistics, multiple regression methods are used to fit the parameters of these polynomials.
The polynomials, along with the fitted coefficients, form the multiple regression equations.
