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. Coordinate conversion comprises a number of different types of conversion: format change of geographic coordinates, conversion of coordinate systems, or transformation to different geodetic datums. Geographic coordinate conversion has applications in cartography, surveying, navigation and geographic information systems.
Contents
- Coordinate format conversion
- Coordinate system conversion
- From geodetic to ECEF coordinates
- From ECEF to geodetic coordinates
- NewtonRaphson method
- Ferraris solution
- Geodetic tofrom ENU coordinates
- From ECEF to ENU
- From ENU to ECEF
- Conversion across map projections
- Datum transformations
- Grid based method
- Molodensky transformation
- Multiple regression equations
- Helmert transformation
- Molodensky Badekas transformation
- References
In geodesy, geographic coordinate conversion is defined as translation among different coordinate formats or map projections all referenced to the same geodetic datum. A geographic coordinate transformation is a translation among different geodetic datums. Both geographic coordinate conversion and transformation will be considered in this article.
This article assumes readers are already familiar with the content in the articles geographic coordinate system and geodetic datum.
Coordinate format conversion
Informally, specifying a geographic location usually means giving the location's latitude and longitude. The numerical values for latitude and longitude can occur in a number of different formats:
There are 60 minutes in a degree and 60 seconds in a minute. Therefore, to convert from a degrees minutes seconds format to a decimal degrees format, one may use the formula
To convert back from decimal degree format to degrees minutes seconds format,
where the notation
Coordinate system conversion
A coordinate system conversion is a conversion from one coordinate system to another, with both coordinate systems based on the same geodetic datum. Common conversion tasks include conversion between geodetic and ECEF coordinates and conversion from one type of map projection to another.
From geodetic to ECEF coordinates
Geodetic coordinates (latitude
where
and
The following equation holds:
where
The orthogonality of the coordinates is confirmed via differentiation:
where
(see also "Meridian arc on the ellipsoid").
From ECEF to geodetic coordinates
The conversion of ECEF coordinates to geodetic coordinates (such WGS84) involves more trigonometry but is sensitive to small accuracy due to Rn and h being maybe 10^6 apart, but longitude is same as geocentric,
There are several methods that solve the equation; two are shown.
Newton–Raphson method
The following Bowring's irrational geodetic-latitude equation is efficient to be solved by Newton–Raphson iteration method:
where
The iteration can be transformed into the following calculation:
where
The constant
Ferrari's solution
The quartic equation of
The application of Ferrari's solution
A number of techniques and algorithms are available but the most accurate according to Zhu, is the following 15 step procedure summarised by Kaplan. It is assumed that geodetic parameters
Note: arctan2[Y,X] is the four-quadrant inverse tangent function.
Geodetic to/from ENU coordinates
To convert from geodetic coordinates to local ENU coordinates is a two-stage process:
- Convert geodetic coordinates to ECEF coordinates
- Convert ECEF coordinates to local ENU coordinates
To convert from local ENU coordinates to geodetic coordinates is a two-stage process
- Convert local ENU coordinates to ECEF coordinates
- Convert ECEF coordinates to geodetic coordinates
From ECEF to ENU
To transform from ECEF coordinates to the local coordinates we need a local reference point, typically this might be the location of a radar. If a radar is located at
Note:
Obtaining geodetic latitude from geocentric coordinates from this relationship requires an iterative solution approach, otherwise the geodetic coordinates may be computed via the approach in the section above labeled "From ECEF to geodetic coordinates."
The geocentric and geodetic longitude have the same value. This is true for the Earth and other similar shaped planets because their latitude lines (parallels) can be considered in much more degree perfect circles when compared to their longitude lines (meridians).
Note: Unambiguous determination of
From ENU to ECEF
This is just the inversion of the ECEF to ENU transformation so
Conversion across map projections
Conversion of coordinates and map positions among different map projections reference to the same datum may be accomplished either through direct translation formulas from one projection to another, or by first converting from a projection
Datum transformations
Transformations among datums can be accomplished in a number of ways. There are transformations that directly convert geodetic coordinates from one datum to another. There are more indirect transforms that convert from geodetic coordinates to ECEF coordinates, transform the ECEF coordinates from one datum to the another, then transform ECEF coordinates of the new datum back to geodetic coordinates. There are also grid-based transformations that directly transform from one (datum, map projection) pair to another (datum, map projection) pair.
Grid-based method
Grid-based transformations directly convert map coordinates from one (map-projection, geodetic datum) pair to map coordinates of another (map-projection, geodetic datum) pair. An example is the NADCON method for transforming from the North American Datum (NAD) 1927 to the NAD 1983 datum. The High Accuracy Reference Network (HARN), a high accuracy version of the NADCON transforms, have an accuracy of approximately 5 centimeters. The National Transformation version 2 (NTv2) is a Canadian version of NADCON for transforming between NAD 1927 and NAD 1983. HARNs are also known as NAD 83/91 and High Precision Grid Networks (HPGN). Subsequently, Australia and New Zealand adopted the NTv2 format to create grid-based methods for transforming among their own local datums.
Like the multiple regression equation transform, grid-based methods use a low-order interpolation method for converting map coordinates, but in two dimensions instead of three. The NOAA provides a software tool (as part of the NGS Geodetic Toolkit) for performing NADCON transformations.
Molodensky transformation
The Molodensky transformation converts directly between geodetic coordinate systems of different datums without the intermediate step of converting to geocentric ECEF coordinates. It requires the three shifts between the datum centers and the differences between the reference ellipsoid semi-major axes and flattening parameters.
The Molodensky transform is used by the National Geospatial-Intelligence Agency (NGA) in their standard TR8350.2 and the NGA supported GEOTRANS program. The Molodensky method was popular before the advent of modern computers and the method is part of many geodetic programs.
Multiple regression equations
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. MRE transforms are used to transform local datums over continent-sized or smaller regions to global datums, such as WGS 84. The standard NIMA TM 8350.2, Appendix D, lists MRE transforms from several local datums to WGS 84, with accuracies of about 2 meters.
The MREs are a direct transformation of geodetic coordinates with no intermediate ECEF step. Geodetic coordinates
where
with similar equations for
Helmert transformation
Use of the Helmert transform in the transformation from geodetic coordinates of datum
- Convert from geodetic coordinates to ECEF coordinates for datum
A - Apply the Helmert transform, with the appropriate
A → B transform parameters, to transform from datumA ECEF coordinates to datumB ECEF coordinates - Convert from ECEF coordinates to geodetic coordinates for datum
B
In terms of ECEF XYZ vectors, the Helmert transform has the form
The Helmert transform is a seven-parameter transform with three translation (shift) parameters
A fourteen-parameter Helmert transform, with linear time dependence for each parameter,:131-133 can be used to capture the time evolution of geographic coordinates dues to geomorphic processes, such as continental drift. and earthquakes. This has been incorporated into software, such as the Horizontal Time Dependent Positioning (HTDP) tool from the U.S. NGS.
Molodensky-Badekas transformation
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. This ten-parameter model is called the Molodensky-Badekas transformation and should not be confused with the more basic Molodensky transform.:133-134
Like the Helmert transform, using the Molodensky-Badekas transform is a three-step process:
- Convert from geodetic coordinates to ECEF coordinates for datum
A - Apply the Molodensky-Badekas transform, with the appropriate
A → B transform parameters, to transform from datumA ECEF coordinates to datumB ECEF coordinates - Convert from ECEF coordinates to geodetic coordinates for datum
B
The transform has the form
where
The Molodensky-Badekas transform is used to transform local geodetic datums to a global geodetic datum, such as WGS 84. Unlike the Helmert transform, the Molodensky-Badekas transform is not reversible due to the rotational origin being associated with the original datum.:134