# Azimuth

An **azimuth** (/ˈæzəməθ/ ^{ⓘ}; from Arabic: اَلسُّمُوت, romanized: *as-sumūt*, lit. 'the directions')^{[1]} is the angular measurement in a spherical coordinate system which represents the horizontal angle from a cardinal direction, most commonly north.

Mathematically, the relative position vector from an observer (origin) to a point of interest is projected perpendicularly onto a reference plane (the horizontal plane); the angle between the projected vector and a reference vector on the reference plane is called the azimuth.

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 with a 5 km 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.^{[2]}

Azimuth is usually measured in degrees (°), in the positive range 0° to 360° or in the signed range -180° to +180°. The concept is used in navigation, astronomy, engineering, mapping, mining, and ballistics.

## Etymology

The word azimuth is used in all European languages today. It originates from medieval Arabic السموت (*al-sumūt*, pronounced *as-sumūt*), meaning "the directions" (plural of Arabic السمت *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. Its first recorded use in English is in the 1390s in Geoffrey Chaucer's *Treatise on the Astrolabe*. 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.^{[3]}

## In astronomy

Main article: Horizontal coordinate system |

In the horizontal coordinate system, used in celestial navigation, azimuth is one of the two coordinates.^{[4]} The other is *altitude*, sometimes called elevation above the horizon.
It is also used for satellite dish installation (see also: sat finder).
In modern astronomy azimuth is nearly always measured from the north.

## In geodesy

Main article: Inverse geodetic problem |

See also: Earth section paths § Inverse problem, Vincenty's formulae § Inverse problem, and Geographical distance § Ellipsoidal-surface formulae |

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** (or **geodesic azimuth**) is the angle between north and the ellipsoidal geodesic (the shortest path on the surface of the spheroid from our viewpoint to Point 2). The difference is usually negligible: less than 0.03 arc second for distances less than 100 km.^{[10]}

Normal-section azimuth can be calculated as follows:^{[citation needed]}

where *f* is the flattening and *e* the eccentricity for the chosen spheroid (e.g., 1⁄298.257223563 for WGS84).
If *φ*_{1} = 0 then

To calculate the azimuth of the Sun or a star given its declination and hour angle at a specific location, 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).^{[citation needed]}

## In cartography

The **cartographical azimuth** or **grid 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.

The opposite problem occurs 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.

### Map projections

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.

## Related coordinates

### Right ascension

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.

### Polar coordinate

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. A special case of an azimuth angle is the angle in polar coordinates of the component of the vector in the *xy*-plane, although this angle is normally measured in radians rather than degrees and denoted by *θ* rather than *φ*.

## Other uses

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.

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