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An **azimuth** ( /ˈæzɪməθ/ (*as-sumūt*, 'the directions', the plural form of the Arabic noun السَّمْت *as-samt*, meaning 'the direction') is an angular measurement in a spherical coordinate system. The vector from an observer (origin) to a point of interest is projected perpendicularly onto a reference plane; the angle between the projected vector and a reference vector on the reference plane is called the azimuth.

- Navigation
- True north-based azimuths
- Cartographical azimuth
- Calculating coordinates
- Calculating azimuth
- Mapping
- Astronomy
- Other systems
- Right ascension
- Horizontal coordinate
- Polar coordinate
- Other uses of the word
- Etymology of the word
- See also
- Notes
- References
- External links

When used as a celestial coordinate, the azimuth is the horizontal direction of a star or other astronomical object in the sky. The star is the point of interest, the reference plane is the local area (e.g. a circular area 5 km in radius at sea level) around an observer on Earth's surface, and the reference vector points to true north. The azimuth is the angle between the north vector and the star's vector on the horizontal plane.^{ [1] }

Azimuth is usually measured in degrees (°). The concept is used in navigation, astronomy, engineering, mapping, mining, and ballistics.

In land navigation, azimuth is usually denoted alpha, *α*, and defined as a horizontal angle measured clockwise from a north base line or * meridian *.^{ [2] }^{ [3] }*Azimuth* has also been more generally defined as a horizontal angle measured clockwise from any fixed reference plane or easily established base direction line.^{ [4] }^{ [5] }^{ [6] }

Today, the reference plane for an azimuth is typically true north, measured as a 0° azimuth, though other angular units (grad, mil) can be used. Moving clockwise on a 360 degree circle, east has azimuth 90°, south 180°, and west 270°. There are exceptions: some navigation systems use south as the reference vector. Any direction can be the reference vector, as long as it is clearly defined.

Quite commonly, azimuths or compass bearings are stated in a system in which either north or south can be the zero, and the angle may be measured clockwise or anticlockwise from the zero. For example, a bearing might be described as "(from) south, (turn) thirty degrees (toward the) east" (the words in brackets are usually omitted), abbreviated "S30°E", which is the bearing 30 degrees in the eastward direction from south, i.e. the bearing 150 degrees clockwise from north. The reference direction, stated first, is always north or south, and the turning direction, stated last, is east or west. The directions are chosen so that the angle, stated between them, is positive, between zero and 90 degrees. If the bearing happens to be exactly in the direction of one of the cardinal points, a different notation, e.g. "due east", is used instead.

North | 0° | South | 180° | |

North-northeast | 22.5° | South-southwest | 202.5° | |

Northeast | 45° | Southwest | 225° | |

East-northeast | 67.5° | West-southwest | 247.5° | |

East | 90° | West | 270° | |

East-southeast | 112.5° | West-northwest | 292.5° | |

Southeast | 135° | Northwest | 315° | |

South-southeast | 157.5° | North-northwest | 337.5° |

The cartographical azimuth (in decimal degrees) can be calculated when the coordinates of 2 points are known in a flat plane (cartographical coordinates):

Remark that the reference axes are swapped relative to the (counterclockwise) mathematical polar coordinate system and that the azimuth is clockwise relative to the north. This is the reason why the X and Y axis in the above formula are swapped. If the azimuth becomes negative, one can always add 360°.

The formula in radians would be slightly easier:

Note the swapped in contrast to the normal atan2 input order.

When the coordinates (*X*_{1}, *Y*_{1}) of one point, the distance *D*, and the azimuth *α* to another point (*X*_{2}, *Y*_{2}) are known, one can calculate its coordinates:

This is typically used in triangulation and azimuth identification (AzID), especially in radar applications.

We are standing at latitude , longitude zero; we want to find the azimuth from our viewpoint to Point 2 at latitude , longitude *L* (positive eastward). We can get a fair approximation by assuming the Earth is a sphere, in which case the azimuth *α* is given by

A better approximation assumes the Earth is a slightly-squashed sphere (an * oblate spheroid *); *azimuth* then has at least two very slightly different meanings. *Normal-section azimuth* is the angle measured at our viewpoint by a theodolite whose axis is perpendicular to the surface of the spheroid; *geodetic azimuth* is the angle between north and the *geodesic*; that is, the shortest path on the surface of the spheroid from our viewpoint to Point 2. The difference is usually immeasurably small; if Point 2 is not more than 100 km away, the difference will not exceed 0.03 arc second.

Various websites will calculate geodetic azimuth; e.g., GeoScience Australia site. Formulas for calculating geodetic azimuth are linked in the distance article.

Normal-section azimuth is simpler to calculate; Bomford says Cunningham's formula is exact for any distance.^{[ citation needed ]} If *f* is the flattening, and *e* the eccentricity, for the chosen spheroid (e.g., ^{1}⁄_{298.257223563} for WGS84) then

If *φ*_{1} = 0 then

To calculate the azimuth of the sun or a star given its declination and hour angle at our location, we modify the formula for a spherical earth. Replace *φ*_{2} with declination and longitude difference with hour angle, and change the sign (since the hour angle is positive westward instead of east).

There is a wide variety of azimuthal map projections. They all have the property that directions (the azimuths) from a central point are preserved. Some navigation systems use south as the reference plane. However, any direction can serve as the plane of reference, as long as it is clearly defined for everyone using that system.

Used in celestial navigation, an *azimuth* is the direction of a celestial body from the observer.^{ [7] } In astronomy, an *azimuth* is sometimes referred to as a bearing. In modern astronomy azimuth is nearly always measured from the north. (The article on coordinate systems, for example, uses a convention measuring from the south.) In former times, it was common to refer to azimuth from the south, as it was then zero at the same time that the hour angle of a star was zero. This assumes, however, that the star (upper) culminates in the south, which is only true if the star's declination is less than (i.e. further south than) the observer's latitude.

If, instead of measuring from and along the horizon, the angles are measured from and along the celestial equator, the angles are called right ascension if referenced to the Vernal Equinox, or hour angle if referenced to the celestial meridian.

In the horizontal coordinate system, used in celestial navigation and satellite dish installation, azimuth is one of the two coordinates. The other is altitude, sometimes called elevation above the horizon. See also: Sat finder.

In mathematics, the azimuth angle of a point in cylindrical coordinates or spherical coordinates is the anticlockwise angle between the positive *x*-axis and the projection of the vector onto the *xy*-plane. The angle is the same as an angle in polar coordinates of the component of the vector in the *xy*-plane and is normally measured in radians rather than degrees. As well as measuring the angle differently, in mathematical applications theta, *θ*, is very often used to represent the azimuth rather than the representation of symbol phi *φ*.

For magnetic tape drives, *azimuth* refers to the angle between the tape head(s) and tape.

In sound localization experiments and literature, the *azimuth* refers to the angle the sound source makes compared to the imaginary straight line that is drawn from within the head through the area between the eyes.

An azimuth thruster in shipbuilding is a propeller that can be rotated horizontally.

The word azimuth is in all European languages today. It originates from medieval Arabic *al-sumūt*, pronounced *as-sumūt* in Arabic, meaning "the directions" (plural of Arabic *al-samt* = "the direction"). The Arabic word entered late medieval Latin in an astronomy context and in particular in the use of the Arabic version of the astrolabe astronomy instrument. The word's first record in English is in the 1390s in * Treatise on the Astrolabe * by Geoffrey Chaucer. The first known record in any Western language is in Spanish in the 1270s in an astronomy book that was largely derived from Arabic sources, the * Libros del saber de astronomía * commissioned by King Alfonso X of Castile.^{ [8] }

- ↑ "Azimuth".
*Dictionary.com*. - ↑ U.S. Army,
*Map Reading and Land Navigation*, FM 21–26, Headquarters, Dept. of the Army, Washington, D.C. (7 May 1993), ch. 6, p. 2 - ↑ U.S. Army,
*Map Reading and Land Navigation*, FM 21–26, Headquarters, Dept. of the Army, Washington, D.C. (28 March 1956), ch. 3, p. 63 - ↑ U.S. Army, ch. 6 p. 2
- ↑ U.S. Army,
*Advanced Map and Aerial Photograph Reading*, Headquarters, War Department, Washington, D.C. (17 September 1941), pp. 24–25 - ↑ U.S. Army,
*Advanced Map and Aerial Photograph Reading*, Headquarters, War Department, Washington, D.C. (23 December 1944), p. 15 - ↑ Rutstrum, Carl, The Wilderness Route Finder, University of Minnesota Press (2000), ISBN 0-8166-3661-3, p. 194
- ↑ "Azimuth" at
*New English Dictionary on Historical Principles*; "azimut" at*Centre National de Ressources Textuelles et Lexicales*; "al-Samt" at*Brill's Encyclopedia of Islam*; "azimuth" at EnglishWordsOfArabicAncestry.wordpress.com Archived January 2, 2014, at the Wayback Machine . In Arabic the written*al-sumūt*is always pronounced*as-sumūt*(see pronunciation of "al-" in Arabic).

The **Mercator projection** is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because of its unique property of representing any course of constant bearing as a straight segment. Such a course, known as a rhumb or, mathematically, a loxodrome, is preferred by navigators because the ship can sail in a constant compass direction to reach its destination, eliminating difficult and error-prone course corrections. Linear scale is constant on the Mercator in every direction around any point, thus preserving the angles and the shapes of small objects and fulfilling the conditions of a conformal map projection. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation starts infinitesimally, but accelerates with latitude to become infinite at the poles. So, for example, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator, such as Central Africa.

In mathematics, the **polar coordinate system** is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the *pole*, and the ray from the pole in the reference direction is the *polar axis*. The distance from the pole is called the *radial coordinate*, *radial distance* or simply *radius*, and the angle is called the *angular coordinate*, *polar angle*, or *azimuth*. The radial coordinate is often denoted by *r* or *ρ*, and the angular coordinate by *φ*, *θ*, or *t*. Angles in polar notation are generally expressed in either degrees or radians.

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 electrodynamics, **elliptical polarization** is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. An elliptically polarized wave may be resolved into two linearly polarized waves in phase quadrature, with their polarization planes at right angles to each other. Since the electric field can rotate clockwise or counterclockwise as it propagates, elliptically polarized waves exhibit chirality.

A **logarithmic spiral**, **equiangular spiral**, or **growth spiral** is a self-similar spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it *Spira mirabilis*, "the marvelous spiral".

In astronomy, a **celestial coordinate system** is a system for specifying positions of satellites, planets, stars, galaxies, and other celestial objects. Coordinate systems can specify an object's position in three-dimensional space or plot merely its direction on a celestial sphere, if the object's distance is unknown or trivial.

In mathematics, a **unit vector** in a normed vector space is a vector of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat": . The term direction vector is used to describe a unit vector being used to represent spatial direction, and such quantities are commonly denoted as **d**. Two 2D direction vectors, **d1** and **d2** are illustrated. 2D spatial directions represented this way are numerically equivalent to points on the unit circle.

In mechanics and geometry, the **3D rotation group**, often denoted **SO(3)**, is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance, and orientation. Every non-trivial rotation is determined by its axis of rotation and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Rotations are not commutative, making it a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable; so it is in fact a Lie group. It is compact and has dimension 3.

A **cylindrical coordinate system** is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point.

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 **Euler angles** are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system. They can also represent the orientation of a mobile frame of reference in physics or the orientation of a general basis in 3-dimensional linear algebra.

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.

**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.

In geometry, various **formalisms** exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

The main **trigonometric identities** between trigonometric functions are proved, using mainly the geometry of the right triangle. For greater and negative angles, see Trigonometric functions.

In mathematics, a **mean of circular quantities** is a mean which is sometimes better-suited for quantities like angles, daytimes, and fractional parts of real numbers. This is necessary since most of the usual means may not be appropriate on circular quantities. For example, the arithmetic mean of 0° and 360° is 180°, which is misleading because for most purposes 360° is the same thing as 0°. As another example, the "average time" between 11 PM and 1 AM is either midnight or noon, depending on whether the two times are part of a single night or part of a single calendar day. This is one of the simplest examples of statistics of non-Euclidean spaces.

In spherical astronomy, the **parallactic angle** is the angle between the great circle through a celestial object and the zenith, and the hour circle of the object. It is usually denoted *q*. In the triangle zenith—object—celestial pole, the parallactic angle will be the position angle of the zenith at the celestial object. Despite its name, this angle is unrelated with parallax. The parallactic angle is zero or 180ˆwhen the object crosses the meridian.

The study of **geodesics on an ellipsoid** arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an *oblate ellipsoid*, a slightly flattened sphere. A *geodesic* is the shortest path between two points on a curved surface, analogous to a straight line on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry.

**Earth section paths** are paths on the earth defined by the intersection of a reference ellipsoid and a plane. Common examples of earth sections include the great ellipse and normal sections. This page provides a unifying approach to all earth sections and their associated geodetic problems.

- Rutstrum, Carl,
*The Wilderness Route Finder*, University of Minnesota Press (2000), ISBN 0-8166-3661-3 - U.S. Army,
*Advanced Map and Aerial Photograph Reading*, FM 21–26, Headquarters, War Department, Washington, D.C. (17 September 1941) - U.S. Army,
*Advanced Map and Aerial Photograph Reading*, FM 21–26, Headquarters, War Department, Washington, D.C. (23 December 1944) - U.S. Army,
*Map Reading and Land Navigation*, FM 21–26, Headquarters, Dept. of the Army, Washington, D.C. (7 May 1993)

Look up in Wiktionary, the free dictionary. azimuth |

*Encyclopædia Britannica*(11th ed.). 1911. . *Collier's New Encyclopedia*. 1921.

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