Geodesy

Fundamentals


Concepts


Technologies


Standards

ED50

European Datum 1950

SAD69

South American Datum 1969

GRS 80

Geodetic Reference System 1980

NAD83

North American Datum 1983

WGS84

World Geodetic System 1984

NAVD88

N. American Vertical Datum 1988

ETRS89

European Terrestrial Reference
System 1989


History



A position representation is the parameters used to express a position relative to a reference. Representing position in three dimensions is often done by a Euclidean vector. However, when representing position relative to the Earth it is often more convenient to represent vertical position as altitude or depth, and to use some other parameters to represent horizontal position. There are also several applications where only the horizontal position is of interest, this might e.g. be the case for ships and ground vehicles/cars.
There are several options for horizontal position representations, each with different properties which makes them appropriate for different applications. Latitude/longitude and UTM are common horizontal position representations.
The horizontal position has two degrees of freedom, and thus two parameters are sufficient to uniquely describe such a position. However, similarly to the use of Euler angles as a formalism for representing rotations, using only the minimum number of parameters gives singularities, and thus three parameters are required for the horizontal position to avoid this.
Latitude and longitude
The most common horizontal position representation is latitude and longitude. The parameters are intuitive and well known, and are thus suited for communicating a position to humans, e.g. using a position plot.
However, latitude and longitude should be used with care in mathematical expressions (including calculations in computer programs). The main reason is the singularities at the Poles, which makes longitude undefined at these points. Also near the poles the latitude/longitude grid is highly nonlinear, and several errors may occur in calculations that are sufficiently accurate on other locations. ^{[1]}
Another problematic area is the meridian at ±180° longitude, where the longitude has a discontinuity, and hence specific program code must often be written to handle this. An example of the consequences of omitting such code is the crash of the navigation systems of twelve F22 Raptors while crossing this meridian.^{[2]}
nvector
nvector is a three parameter nonsingular horizontal position representation that can replace latitude and longitude. Geometrically, it is a unit vector which is normal to the reference ellipsoid. The vector is decomposed in an Earth centered earth fixed coordinate system. It behaves the same at all Earth positions, and it holds the mathematical onetoone property. The vector formulation makes it possible to use standard 3D vector algebra, and thus nvector is wellsuited for mathematical calculations, e.g. adding, subtracting, interpolating and averaging positions.
Using three parameters, nvector is inconvenient for communicating a position directly to humans and before showing a position plot, a conversion to latitude/longitude might be needed.
Local flat Earth assumption
When carrying out several calculations within a limited area, a Cartesian coordinate system might be defined with the origin at a specified Earthfixed position. The origin is often selected at the surface of the reference ellipsoid, with the zaxis in the vertical direction. Hence (three dimensional) position vectors relative to this coordinate frame will have two horizontal and one vertical parameter. The axes are typically selected as NorthEastDown or EastNorthUp, and thus this system can be viewed as a linearization of the meridians and parallels.
For small areas a local coordinate system can be convenient for relative positioning, but with increasing (horizontal) distances, errors will increase and repositioning of the tangent point may be required. The alignment along the north and east directions is not possible at the Poles, and near the Poles these directions might have significant errors (here the linearization is valid only in a very small area).
UTM
Instead of one local Cartesian grid, that needs to be repositioned as the position of interest moves, a fixed set of map projections covering the Earth can be defined. UTM is one such system, dividing the Earth into 60 longitude zones (and with UPS covering the Polar regions).
UTM is widely used, and the coordinates approximately corresponds to meters north and east. However, as a set of mapprojections it has inherent distortions, and thus most calculations based on UTM will not be exact. The crossing of zones gives additional complexity.
Comparison
When deciding which parameters to use for representing position in a specific application, there are several properties that should be considered. If separate horizontal and vertical parameters are not needed, a Cartesian vector in the ECEF coordinate system might be a good choice. For applications where separate horizontal and vertical parameters are most convenient, the following table gives a summary of what to consider.
Comparison of horizontal position representations
Representation

Pros

Cons

Latitude and longitude


Widely used

Parameters are easy to recognize by humans (wellsuited for plotting)


nvector


Nonsingular

Efficient in equations/calculations since standard 3D vector algebra can be used

All Earth positions are treated equally


Inconvenient for communicating a position to humans

Uses three parameters

Local Cartesian coordinate system


Cartesian vectors in meters along the directions of north, east and down are obtained


Can only be used for relative positioning (the tangent point must be represented by some other quantity)

Errors increase with increasing horizontal distance from the tangent point (which may require repositioning of the tangent point)

North and east directions are undefined at the Poles, and near the Poles these directions may change significantly within the area of interest

UTM


Widely used

Approximate north and east directions

One unit corresponds approximately to one meter


See also
References

^ Gade, Kenneth (2010). "A nonsingular horizontal position representation" (PDF). The Journal of Navigation (Cambridge University Press) 63 (3): 395–417.

^ "Stealth fighters hit by software crash". 26 February 2007. Retrieved 7 June 2010.
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