Horizontal position representation

Last updated September 22, 2023

A position representation is a set of parameters used to express a position relative to a reference. 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.

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 non-linear, 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 F-22 Raptor fighter aircraft while crossing this meridian.^[2]

n-vector

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.

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 Earth-fixed position. The origin is often selected at the surface of the reference ellipsoid, with the z-axis 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 North-East-Down or East-North-Up, 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).

Grid reference system

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

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 map-projections 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. 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 (well-suited for plotting)	Singularities at the Poles Complex behavior near the Poles Discontinuity at the ±180° meridian
n-vector	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	Inherent distortion (due to the map projection) gives only approximate answers for most calculations Calculations get complex when crossing the zones The Polar Regions are not covered

Related Research Articles

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<span class="mw-page-title-main">Geodesy</span> Science of measuring the shape, orientation, and gravity of the Earth and other astronomical bodies

Geodesy is the science of measuring and representing the geometry, gravity, and spatial orientation of the Earth in temporally varying 3D. It is called planetary geodesy when studying other astronomical bodies, such as planets or circumplanetary systems.

In geography, latitude is a coordinate that specifies the north–south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pole, with 0° at the Equator. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude and longitude are used together as a coordinate pair to specify a location on the surface of the Earth.

Longitude is a geographic coordinate that specifies the east–west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter lambda (λ). Meridians are imaginary semicircular lines running from pole to pole that connect points with the same longitude. The prime meridian defines 0° longitude; by convention the International Reference Meridian for the Earth passes near the Royal Observatory in Greenwich, south-east London on the island of Great Britain. Positive longitudes are east of the prime meridian, and negative ones are west.

An azimuth is the angular measurement in a spherical coordinate system which represents the horizontal angle from a cardinal direction, most commonly north.

<span class="mw-page-title-main">Geographic coordinate system</span> System to specify locations on Earth

The geographic coordinate system (GCS) is a spherical or geodetic coordinate system for measuring and communicating positions directly on the Earth as latitude and longitude. It is the simplest, oldest and most widely used of the various spatial reference systems that are in use, and forms the basis for most others. Although latitude and longitude form a coordinate tuple like a cartesian coordinate system, the geographic coordinate system is not cartesian because the measurements are angles and are not on a planar surface.

<span class="mw-page-title-main">Projected coordinate system</span> Cartesian geographic coordinate system

A projected coordinate system – also called a projected coordinate reference system, planar coordinate system, or grid reference system – is a type of spatial reference system that represents locations on Earth using Cartesian coordinates (x, y) on a planar surface created by a particular map projection. Each projected coordinate system, such as "Universal Transverse Mercator WGS 84 Zone 26N," is defined by a choice of map projection (with specific parameters), a choice of geodetic datum to bind the coordinate system to real locations on the earth, an origin point, and a choice of unit of measure. Hundreds of projected coordinate systems have been specified for various purposes in various regions.

<span class="mw-page-title-main">World Geodetic System</span> Geodetic reference system

The World Geodetic System (WGS) is a standard used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also describes the associated Earth Gravitational Model (EGM) and World Magnetic Model (WMM). The standard is published and maintained by the United States National Geospatial-Intelligence Agency.

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The transverse Mercator map projection is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the Universal Transverse Mercator. When paired with a suitable geodetic datum, the transverse Mercator delivers high accuracy in zones less than a few degrees in east-west extent.

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<span class="mw-page-title-main">Geodetic datum</span> Reference frame for measuring location

A geodetic datum or geodetic system is a global datum reference or reference frame for precisely representing the position of locations on Earth or other planetary bodies by means of geodetic coordinates. Datums are crucial to any technology or technique based on spatial location, including geodesy, navigation, surveying, geographic information systems, remote sensing, and cartography. A horizontal datum is used to measure a location across the Earth's surface, in latitude and longitude or another coordinate system; a vertical datum is used to measure the elevation or depth relative to a standard origin, such as mean sea level (MSL). Since the rise of the global positioning system (GPS), the ellipsoid and datum WGS 84 it uses has supplanted most others in many applications. The WGS 84 is intended for global use, unlike most earlier datums.

<span class="mw-page-title-main">Universal Transverse Mercator coordinate system</span> System for assigning planar coordinates to locations on the surface of the Earth.

The Universal Transverse Mercator (UTM) is a map projection system for assigning coordinates to locations on the surface of the Earth. Like the traditional method of latitude and longitude, it is a horizontal position representation, which means it ignores altitude and treats the earth surface as a perfect ellipsoid. However, it differs from global latitude/longitude in that it divides earth into 60 zones and projects each to the plane as a basis for its coordinates. Specifying a location means specifying the zone and the x, y coordinate in that plane. The projection from spheroid to a UTM zone is some parameterization of the transverse Mercator projection. The parameters vary by nation or region or mapping system.

<span class="mw-page-title-main">Spatial reference system</span> System to specify locations on Earth

A spatial reference system (SRS) or coordinate reference system (CRS) is a framework used to precisely measure locations on the surface of Earth as coordinates. It is thus the application of the abstract mathematics of coordinate systems and analytic geometry to geographic space. A particular SRS specification comprises a choice of Earth ellipsoid, horizontal datum, map projection, origin point, and unit of measure. Thousands of coordinate systems have been specified for use around the world or in specific regions and for various purposes, necessitating transformations between different SRS.

The n-vector representation is a three-parameter non-singular representation well-suited for replacing geodetic coordinates for horizontal position representation in mathematical calculations and computer algorithms.

The Earth-centered, Earth-fixed coordinate system, also known as the geocentric coordinate system, is a cartesian spatial reference system that represents locations in the vicinity of the Earth as X, Y, and Z measurements from its center of mass. Its most common use is in tracking the orbits of satellites and in satellite navigation systems for measuring locations on the surface of the Earth, but it is also used in applications such as tracking crustal motion.

Local tangent plane coordinates (LTP), also known as local ellipsoidal system, local geodetic coordinate system, or local vertical, local horizontal coordinates (LVLH), are a spatial reference system based on the tangent plane defined by the local vertical direction and the Earth's axis of rotation. It consists of three coordinates: one represents the position along the northern axis, one along the local eastern axis, and one represents the vertical position. Two right-handed variants exist: east, north, up (ENU) coordinates and north, east, down (NED) coordinates. They serve for representing state vectors that are commonly used in aviation and marine cybernetics.

<span class="mw-page-title-main">Earth ellipsoid</span> Shape of planet Earth

An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations.

Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a reference ellipsoid. They include geodetic latitude (north/south) $ϕ$ , longitude (east/west) $λ$ , and ellipsoidal height $h$ . The triad is also known as Earth ellipsoidal coordinates.

A planetary coordinate system is a generalization of the geographic, geodetic, and the geocentric coordinate systems for planets other than Earth. Similar coordinate systems are defined for other solid celestial bodies, such as in the selenographic coordinates for the Moon. The coordinate systems for almost all of the solid bodies in the Solar System were established by Merton E. Davies of the Rand Corporation, including Mercury, Venus, Mars, the four Galilean moons of Jupiter, and Triton, the largest moon of Neptune.

References

↑ Gade, Kenneth (2010). "A non-singular horizontal position representation" (PDF). The Journal of Navigation. Cambridge University Press. 63 (3): 395–417. Bibcode:2010JNav...63..395G. doi:10.1017/S0373463309990415.
↑ "Stealth fighters hit by software crash". 27 February 2007.

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[1] Gade, Kenneth (2010). "A non-singular horizontal position representation" (PDF). The Journal of Navigation. Cambridge University Press. 63 (3): 395–417. Bibcode:2010JNav...63..395G. doi:10.1017/S0373463309990415.

[2] "Stealth fighters hit by software crash". 27 February 2007.

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