The **azimuthal equidistant projection** is an azimuthal map projection. It has the useful properties that all points on the map are at proportionally correct distances from the center point, and that all points on the map are at the correct azimuth (direction) from the center point. A useful application for this type of projection is a polar projection which shows all meridians (lines of longitude) as straight, with distances from the pole represented correctly. The flag of the United Nations contains an example of a polar azimuthal equidistant projection.

While it may have been used by ancient Egyptians for star maps in some holy books,^{ [1] } the earliest text describing the azimuthal equidistant projection is an 11th-century work by al-Biruni.^{ [2] }

The projection appears in many Renaissance maps, and Gerardus Mercator used it for an inset of the north polar regions in sheet 13 and legend 6 of his well-known 1569 map. In France and Russia this projection is named "Postel projection" after Guillaume Postel, who used it for a map in 1581.^{ [3] } Many modern star chart planispheres use the polar azimuthal equidistant projection.

A point on the globe is chosen as "the center" in the sense that mapped distances and azimuth directions from that point to any other point will be correct. That point, (*φ*_{1}, *λ*_{0}), will project to the center of a circular projection, with *φ* referring to latitude and *λ* referring to longitude. All points along a given azimuth will project along a straight line from the center, and the angle *θ* that the line subtends from the vertical is the azimuth angle. The distance from the center point to another projected point *ρ* is the arc length along a great circle between them on the globe. By this description, then, the point on the plane specified by (*θ*,*ρ*) will be projected to Cartesian coordinates:

The relationship between the coordinates (*θ*,*ρ*) of the point on the globe, and its latitude and longitude coordinates (*φ*, *λ*) is given by the equations: ^{ [4] }

When the center point is the north pole, *φ*_{1} equals and *λ*_{0} is arbitrary, so it is most convenient to assign it the value of 0. This assignment significantly simplifies the equations for *ρ*_{u} and *θ* to:

With the circumference of the Earth being approximately 40,000 km (24,855 mi), the maximum distance that can be displayed on an azimuthal equidistant projection map is half the circumference, or about 20,000 km (12,427 mi). For distances less than 10,000 km (6,214 mi) distortions are minimal. For distances 10,000–15,000 km (6,214–9,321 mi) the distortions are moderate. Distances greater than 15,000 km (9,321 mi) are severely distorted.

If the azimuthal equidistant projection map is centered about a point whose antipodal point lies on land and the map is extended to the maximum distance of 20,000 km (12,427 mi) the antipode point smears into a large circle. This is shown in the example of two maps centered about Los Angeles, and Taipei. The antipode for Los Angeles is in the south Indian Ocean hence there is not much significant distortion of land masses for the Los Angeles centered map except for East Africa and Madagascar. On the other hand, Taipei's antipode is near the Argentina–Paraguay border, causing the Taipei centered map to severely distort South America.

- Map centered about Taipei, whose antipodal point is near the Argentina–Paraguay border.

Brazil · Paraguay · Argentina · Chile · Bolivia · Uruguay

Azimuthal equidistant projection maps can be useful in terrestrial point to point communication. This type of projection allows the operator to easily determine in which direction to point their directional antenna. The operator simply finds on the map the location of the target transmitter or receiver (i.e. the other antenna being communicated with) and uses the map to determine the azimuth angle needed to point the operator's antenna. The operator would use an electric rotator to point the antenna. The map can also be used in one way communication. For example if the operator is looking to receive signals from a distant radio station, this type of projection could help identify the direction of the distant radio station. In order for the map to be useful, the map should be centered as close as possible about the location of the operator's antenna.^{[ citation needed ]}

Azimuthal equidistant projection maps can also be useful to show ranges of missiles, as demonstrated by the map on the right.

In mathematics, a **spherical coordinate system** is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the *radial distance* of that point from a fixed origin, its *polar angle* measured from a fixed zenith direction, and the *azimuthal angle* of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

In mathematics, **Laplace's equation** is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as

An **ellipsoid** is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

In navigation, a **rhumb line**, **rhumb**, or **loxodrome** is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true or magnetic north.

The **haversine formula** determines the great-circle distance between two points on a sphere given their longitudes and latitudes. Important in navigation, it is a special case of a more general formula in spherical trigonometry, the **law of haversines**, that relates the sides and angles of spherical triangles.

The use of **orthographic projection in cartography** dates back to antiquity. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective projection, in which the sphere is projected onto a tangent plane or secant plane. The *point of perspective* for the orthographic projection is at infinite distance. It depicts a hemisphere of the globe as it appears from outer space, where the horizon is a great circle. The shapes and areas are distorted, particularly near the edges.

The **scale** of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways.

The **Bonne projection** is a pseudoconical equal-area map projection, sometimes called a **dépôt de la guerre**, **modified Flamsteed**, or a **Sylvanus** projection. Although named after Rigobert Bonne (1727–1795), the projection was in use prior to his birth, in 1511 by Sylvano, Honter in 1561, De l'Isle before 1700 and Coronelli in 1696. Both Sylvano and Honter's usages were approximate, however, and it is not clear they intended to be the same projection.

The **Bottomley map projection** is an equal area map projection defined as:

In cartography, a **Tissot's indicatrix** is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 and 1871 in order to characterize local distortions due to map projection. It is the geometry that results from projecting a circle of infinitesimal radius from a curved geometric model, such as a globe, onto a map. Tissot proved that the resulting diagram is an ellipse whose axes indicate the two principal directions along which scale is maximal and minimal at that point on the map.

A **Lambert conformal conic projection** (**LCC**) is a conic map projection used for aeronautical charts, portions of the State Plane Coordinate System, and many national and regional mapping systems. It is one of seven projections introduced by Johann Heinrich Lambert in his 1772 publication *Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten*.

The **Albers equal-area conic projection**, or **Albers projection**, is a conic, equal area map projection that uses two standard parallels. Although scale and shape are not preserved, distortion is minimal between the standard parallels.

The **Lambert azimuthal equal-area projection** is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles. It is named for the Swiss mathematician Johann Heinrich Lambert, who announced it in 1772. "Zenithal" being synonymous with "azimuthal", the projection is also known as the **Lambert zenithal equal-area projection**.

**Position angle**, usually abbreviated **PA**, is the convention for measuring angles on the sky in astronomy. The International Astronomical Union defines it as the angle measured relative to the north celestial pole (NCP), turning positive into the direction of the right ascension. In the standard (non-flipped) images this is a counterclockwise measure relative to the axis into the direction of positive declination.

**Cylindrical multipole moments** are the coefficients in a series expansion of a potential that varies logarithmically with the distance to a source, i.e., as . Such potentials arise in the electric potential of long line charges, and the analogous sources for the magnetic potential and gravitational potential.

The **Aitoff projection** is a modified azimuthal map projection proposed by David A. Aitoff in 1889. Based on the equatorial form of the azimuthal equidistant projection, Aitoff first halves longitudes, then projects according to the azimuthal equidistant, and then stretches the result horizontally into a 2:1 ellipse to compensate for having halved the longitudes. Expressed simply:

The **Hammer projection** is an equal-area map projection described by Ernst Hammer in 1892. Using the same 2:1 elliptical outer shape as the Mollweide projection, Hammer intended to reduce distortion in the regions of the outer meridians, where it is extreme in the Mollweide.

In mathematics, the **spectral theory of ordinary differential equations** is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.

The **Eckert IV projection** is an equal-area pseudocylindrical map projection. The length of the polar lines is half that of the equator, and lines of longitude are semiellipses, or portions of ellipses. It was first described by Max Eckert in 1906 as one of a series of three pairs of pseudocylindrical projections. Within each pair, meridians are the same whereas parallels differ. Odd-numbered projections have parallels spaced equally, whereas even-numbered projections have parallels spaced to preserve area. Eckert IV is paired with Eckert III.

The **Hammer retroazimuthal projection** is a modified azimuthal proposed by Ernst Hermann Heinrich Hammer in 1910. As a retroazimuthal projection, azimuths (directions) are correct from any point to the designated center point. Additionally, all distances from the center of the map are proportional to what they are on the globe. In whole-world presentation, the back and front hemispheres overlap, making the projection a non-injective function. The back hemisphere can be rotated 180° to avoid overlap, but in this case, any azimuths measured from the back hemisphere must be corrected.

- ↑ SNYDER, John P. (1997).
*Flattening the earth: two thousand years of map projections*. University of Chicago Press. ISBN 0-226-76747-7., p.29 - ↑ David A. KING (1996), "Astronomy and Islamic society: Qibla, gnomics and timekeeping", in Roshdi Rashed, ed.,
*Encyclopedia of the History of Arabic Science*, Vol. 1, p. 128–184 [153]. Routledge, London and New York. - ↑ Snyder 1997, p. 29
- ↑ Snyder, John P.; Voxland, Philip M. (1989).
*An Album of Map Projections*. Professional Paper 1453. Denver: USGS. p. 228. ISBN 978-0160033681 . Retrieved 2018-03-29.

- Table of examples and properties of all common projections, from radicalcartography.net
- Online Azimuthal Equidistant Map Generator
- An interactive Java Applet to study the metric deformations of the Azimuthal Equidistant Projection.
- GeographicLib provides a class for performing azimuthal equidistant projections centered at any point on the ellipsoid.
- Animated US National Weather Service Wind Data for Azimuthal equidistant projection.
- Generate an Azimuthal equidistant projection from any point on Earth from ns6t.net.

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