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

Earth

Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

 it is often more convenient to represent vertical position as altitude

Altitude

Altitude or height is defined based on the context in which it is used . As a general definition, altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The reference datum also often varies according to the context...

 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

Latitude

In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

/longitude

Longitude

Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

 and UTM

Universal Transverse Mercator coordinate system

The Universal Transverse Mercator geographic coordinate system uses a 2-dimensional Cartesian coordinate system to give locations on the surface of the Earth. It is a horizontal position representation, i.e...

 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 as for rotation representations, using only the minimum number of parameters gives singularities

Mathematical singularity

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...

, and thus three parameters are required for the horizontal position to avoid this.

Latitude and longitude

The most common horizontal position representation is latitude

Latitude

In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

 and longitude

Longitude

Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

. 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

Mathematical singularity

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...

 at the Poles

Geographical pole

A geographical pole is either of the two points—the north pole and the south pole—on the surface of a rotating planet where the axis of rotation meets the surface of the body...

, 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

Meridian (geography)

A meridian is an imaginary line on the Earth's surface from the North Pole to the South Pole that connects all locations along it with a given longitude. The position of a point along the meridian is given by its latitude. Each meridian is perpendicular to all circles of latitude...

 at ±180°

180th meridian

The 180th meridian or antimeridian is the meridian which is 180° east or west of the Prime Meridian passing through the Royal Observatory, Greenwich. It is common to both east longitude and west longitude. It is used as the basis for the International Date Line because it for the most part passes...

 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

F-22 Raptor

The Lockheed Martin/Boeing F-22 Raptor is a single-seat, twin-engine fifth-generation supermaneuverable fighter aircraft that uses stealth technology. It was designed primarily as an air superiority fighter, but has additional capabilities that include ground attack, electronic warfare, and signals...

s while crossing this meridian.

n-vector

n-vector

N-vector

n-vector is a three parameter non-singular horizontal position representation well-suited for replacing latitude and longitude in mathematical calculations and computer algorithms. Geometrically, it is a unit vector that is normal to the reference ellipsoid. The vector is decomposed in an Earth...

 is a three parameter non-singular

Mathematical singularity

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...

 horizontal position representation that can replace latitude and longitude. Geometrically, it is a unit vector which is normal to the reference ellipsoid

Reference ellipsoid

In geodesy, a reference ellipsoid is a mathematically-defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body....

. The vector is decomposed

Vector decomposition

Vector decomposition refers to decomposing a vector of Rn into several vectors, each linearly independent .-Vector decomposition in two dimensions:...

 in an Earth centered earth fixed

ECEF

ECEF stands for Earth-Centered, Earth-Fixed, and is a Cartesian coordinate system, and is sometimes known as a "conventional terrestrial" system. It represents positions as an X, Y, and Z coordinate. The point is defined as the center of mass of the earth, hence the name Earth-Centered...

 coordinate system

Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

. It behaves the same at all Earth positions, and it holds the mathematical one-to-one

Injective function

In mathematics, an injective function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is mapped to by at most one element of its domain...

 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

Cartesian coordinate system

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

 might be defined with the origin

Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...

 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

North East Down

North east down , also known as local tangent plane , is a geographical coordinate system for representing state vectors that is commonly used in aviation. It consists of three numbers: one represents the position along the northern axis, one along the eastern axis, and one represents vertical...

 or East-North-Up, and thus this system can be viewed as a linearization

Linearization

In mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or...

 of the meridians

Meridian (geography)

A meridian is an imaginary line on the Earth's surface from the North Pole to the South Pole that connects all locations along it with a given longitude. The position of a point along the meridian is given by its latitude. Each meridian is perpendicular to all circles of latitude...

 and parallels

Circle of latitude

A circle of latitude, on the Earth, is an imaginary east-west circle connecting all locations that share a given latitude...

.

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

Geographical pole

A geographical pole is either of the two points—the north pole and the south pole—on the surface of a rotating planet where the axis of rotation meets the surface of the body...

, 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

Universal Transverse Mercator coordinate system

The Universal Transverse Mercator geographic coordinate system uses a 2-dimensional Cartesian coordinate system to give locations on the surface of the Earth. It is a horizontal position representation, i.e...

 is one such system, dividing the Earth into 60 longitude zones (and with UPS

Universal Polar Stereographic coordinate system

The Universal Polar Stereographic coordinate system is used in conjunction with the Universal Transverse Mercator coordinate system to locate positions on the surface of the earth. Like the UTM coordinate system, the UPS coordinate system uses a metric-based cartesian grid laid out on a...

 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.
If separate horizontal and vertical parameters are not needed, a Cartesian vector in the ECEF

ECEF

ECEF stands for Earth-Centered, Earth-Fixed, and is a Cartesian coordinate system, and is sometimes known as a "conventional terrestrial" system. It represents positions as an X, Y, and Z coordinate. The point is defined as the center of mass of the earth, hence the name Earth-Centered...

 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 Latitude In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...  and longitude Longitude Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....	Widely used Parameters are easy to recognize by humans (well-suited for plotting)	Singularities Mathematical singularity In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...  at the Poles Geographical pole A geographical pole is either of the two points—the north pole and the south pole—on the surface of a rotating planet where the axis of rotation meets the surface of the body... Complex behavior near the Poles Discontinuity at the ±180° meridian 180th meridian The 180th meridian or antimeridian is the meridian which is 180° east or west of the Prime Meridian passing through the Royal Observatory, Greenwich. It is common to both east longitude and west longitude. It is used as the basis for the International Date Line because it for the most part passes...
n-vector N-vector n-vector is a three parameter non-singular horizontal position representation well-suited for replacing latitude and longitude in mathematical calculations and computer algorithms. Geometrically, it is a unit vector that is normal to the reference ellipsoid. The vector is decomposed in an Earth...	Nonsingular Mathematical singularity In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability... 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 coordinate system A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...	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 Universal Transverse Mercator coordinate system The Universal Transverse Mercator geographic coordinate system uses a 2-dimensional Cartesian coordinate system to give locations on the surface of the Earth. It is a horizontal position representation, i.e...	Widely used Approximate north and east directions One unit corresponds approximately to one meter	Inherent distortion Distortion A distortion is the alteration of the original shape of an object, image, sound, waveform or other form of information or representation. Distortion is usually unwanted, and often many methods are employed to minimize it in practice...  (due to the map projection Map projection A map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane. Map projections are necessary for creating maps. All map projections distort the surface in some fashion... ) gives only approximate answers for most calculations Calculations get complex when crossing the zones The Polar Regions are not covered