Horizontal position representation

A position representation is a set of parameters used to express a position relative to a reference frame. When representing positions relative to the Earth, it is often most convenient to represent vertical position (height or depth) separately, 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. It is a type of geographic coordinate system.

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.

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 non-linear, and several errors may occur in calculations that are sufficiently accurate on other locations.

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 F-22 Raptor fighter aircraft while crossing this meridian.

n-vector is a three parameter non-singular 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 one-to-one property. The vector formulation makes it possible to use standard 3D vector algebra, and thus n-vector is well-suited for mathematical calculations, e.g. adding, subtracting, interpolating and averaging positions.

Using three parameters, n-vector is inconvenient for communicating a position directly to humans and before showing a position plot, a conversion to latitude/longitude might be needed.