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An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an
angular measurement In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles are ...
in a
spherical coordinate system 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'' measu ...
. More specifically, it is the horizontal angle from a
cardinal direction The four cardinal directions, or cardinal points, are the four main compass directions: north, east, south, and west, commonly denoted by their initials N, E, S, and W respectively. Relative to north, the directions east, south, and west are at ...
, most commonly
north North is one of the four compass points or cardinal directions. It is the opposite of south and is perpendicular to east and west. ''North'' is a noun, adjective, or adverb indicating Direction (geometry), direction or geography. Etymology T ...
. Mathematically, the
relative position In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or ...
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
from an observer (
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
) to a point of interest is
projected Projected is an American rock supergroup consisting of Sevendust members John Connolly and Vinnie Hornsby, Alter Bridge and Creed drummer Scott Phillips, and former Submersed and current Tremonti guitarist Eric Friedman. The band released thei ...
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
ly onto a
reference plane In celestial mechanics, the plane of reference (or reference plane) is the plane used to define orbital elements (positions). The two main orbital elements that are measured with respect to the plane of reference are the inclination and the longi ...
(the
horizontal plane In astronomy, geography, and related sciences and contexts, a '' direction'' or ''plane'' passing by a given point is said to be vertical if it contains the local gravity direction at that point. Conversely, a direction or plane is said to be hor ...
); the angle between the projected vector and a reference vector on the reference plane is called the azimuth. When used as a
celestial coordinate Astronomical coordinate systems are organized arrangements for specifying positions of satellites, planets, stars, galaxies, and other celestial objects relative to physical reference points available to a situated observer (e.g. the true horizon ...
, the azimuth is the
horizon The horizon is the apparent line that separates the surface of a celestial body from its sky when viewed from the perspective of an observer on or near the surface of the relevant body. This line divides all viewing directions based on whether i ...
tal direction of a
star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
or other
astronomical object An astronomical object, celestial object, stellar object or heavenly body is a naturally occurring physical entity, association, or structure that exists in the observable universe. In astronomy, the terms ''object'' and ''body'' are often us ...
in the
sky The sky is an unobstructed view upward from the surface of the Earth. It includes the atmosphere and outer space. It may also be considered a place between the ground and outer space, thus distinct from outer space. In the field of astronomy, ...
. 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 Mean sea level (MSL, often shortened to sea level) is an average surface level of one or more among Earth's coastal bodies of water from which heights such as elevation may be measured. The global MSL is a type of vertical datuma standardised g ...
) around an observer on
Earth's surface Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surface ...
, and the reference vector points to
true north True north (also called geodetic north or geographic north) is the direction along Earth's surface towards the geographic North Pole or True North Pole. Geodetic north differs from ''magnetic'' north (the direction a compass points toward the ...
. The azimuth is the angle between the north vector and the star's vector on the
horizontal plane In astronomy, geography, and related sciences and contexts, a '' direction'' or ''plane'' passing by a given point is said to be vertical if it contains the local gravity direction at that point. Conversely, a direction or plane is said to be hor ...
. Azimuth is usually measured in degrees (°). The concept is used in
navigation Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...
,
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
,
engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
,
map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
ping, mining, and
ballistics Ballistics is the field of mechanics concerned with the launching, flight behaviour and impact effects of projectiles, especially ranged weapon munitions such as bullets, unguided bombs, rockets or the like; the science or art of designing and a ...
.


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 An astrolabe ( grc, ἀστρολάβος ; ar, ٱلأَسْطُرلاب ; persian, ستاره‌یاب ) is an ancient astronomical instrument that was a handheld model of the universe. Its various functions also make it an elaborate inclin ...
astronomy instrument. Its first recorded use in English is in the 1390s in
Geoffrey Chaucer Geoffrey Chaucer (; – 25 October 1400) was an English poet, author, and civil servant best known for ''The Canterbury Tales''. He has been called the "father of English literature", or, alternatively, the "father of English poetry". He wa ...
's ''
Treatise on the Astrolabe ''A Treatise on the Astrolabe'' is a medieval instruction manual on the astrolabe by Geoffrey Chaucer. It describes both the form and the proper use of the instrument, and stands out as a prose technical work from a writer better known for poet ...
''. 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 The , ( osp, Libro del saber de astrología), "Books of wisdom of astronomy/astrology of King Alfonso X of Castile", is a series of books of the medieval period, composed during the reign of Alfonso X of Castile. They describe the celestial bodies ...
'' commissioned by
King Alfonso X Alfonso X (also known as the Wise, es, el Sabio; 23 November 1221 – 4 April 1284) was King of Castile, León and Galicia from 30 May 1252 until his death in 1284. During the election of 1257, a dissident faction chose him to be king of Germ ...
of Castile.


In astronomy

In the
horizontal coordinate system The horizontal coordinate system is a celestial coordinate system that uses the observer's local horizon as the fundamental plane to define two angles: altitude and azimuth. Therefore, the horizontal coordinate system is sometimes called as th ...
, used in
celestial navigation Celestial navigation, also known as astronavigation, is the practice of position fixing using stars and other celestial bodies that enables a navigator to accurately determine their actual current physical position in space (or on the surface of ...
, azimuth is one of the two
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
. The other is ''
altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
'', sometimes called elevation above the horizon. It is also used for
satellite dish A satellite dish is a dish-shaped type of parabolic antenna designed to receive or transmit information by radio waves to or from a communication satellite A communications satellite is an artificial satellite that relays and amplifies radi ...
installation (see also:
sat finder A satellite dish is a dish-shaped type of parabolic antenna designed to receive or transmit information by radio waves to or from a communication satellite. The term most commonly means a dish which receives direct-broadcast satellite television ...
). In modern
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
azimuth is nearly always measured from the north.


In navigation

In land navigation, azimuth is usually denoted
alpha Alpha (uppercase , lowercase ; grc, ἄλφα, ''álpha'', or ell, άλφα, álfa) is the first letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter aleph , whic ...
, ''α'', and defined as a horizontal angle measured
clockwise Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...
from a north base line or ''
meridian Meridian or a meridian line (from Latin ''meridies'' via Old French ''meridiane'', meaning “midday”) may refer to Science * Meridian (astronomy), imaginary circle in a plane perpendicular to the planes of the celestial equator and horizon * ...
''. ''Azimuth'' has also been more generally defined as a horizontal angle measured clockwise from any fixed reference plane or easily established base direction line.U.S. Army, ''Advanced Map and Aerial Photograph Reading'', Headquarters, War Department, Washington, D.C. (23 December 1944), p. 15 Today, the reference plane for an azimuth is typically
true north True north (also called geodetic north or geographic north) is the direction along Earth's surface towards the geographic North Pole or True North Pole. Geodetic north differs from ''magnetic'' north (the direction a compass points toward the ...
, 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 point The four cardinal directions, or cardinal points, are the four main compass directions: north, east, south, and west, commonly denoted by their initials N, E, S, and W respectively. Relative to north, the directions east, south, and west are a ...
s, a different notation, e.g. "due east", is used instead.


True north-based azimuths


In geodesy

We are standing at latitude \varphi_1, longitude zero; we want to find the azimuth from our viewpoint to Point 2 at latitude \varphi_2, 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 :\tan\alpha = \frac A better approximation assumes the Earth is a slightly-squashed sphere (an ''
oblate spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circ ...
''); ''azimuth'' then has at least two very slightly different meanings. Normal-section azimuth is the angle measured at our viewpoint by a
theodolite A theodolite () is a precision optical instrument for measuring angles between designated visible points in the horizontal and vertical planes. The traditional use has been for land surveying, but it is also used extensively for building and i ...
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 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 ''geodesi ...
(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.Torge & Müller (2012) Geodesy, De Gruyter, eq.6.70, p.248 Normal-section azimuth can be calculated as follows: : \begin e^2 &= f(2 - f) \\ 1 - e^2 &= (1 - f)^2 \\ \Lambda &= \left(1 - e^2\right) \frac + e^2 \sqrt \\ \tan\alpha &= \frac \end where ''f'' is the flattening and ''e'' the eccentricity for the chosen spheroid (e.g., for
WGS84 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 descr ...
). If ''φ''1 = 0 then : \tan\alpha = \frac 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).


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): :\alpha = \frac \operatorname(X_2 - X_1, Y_2 - Y_1) Remark that the reference axes are swapped relative to the (counterclockwise) mathematical
polar coordinate system 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 (analogous to the or ...
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
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s would be slightly easier: :\alpha = \operatorname(X_2 - X_1, Y_2 - Y_1) Note the swapped (x, y) in contrast to the normal (y, x)
atan2 In computing and mathematics, the function atan2 is the 2-argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi < \theta \leq \pi) between the positive
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: :\begin X_2 &= X_1 + D \sin\alpha \\ Y_2 &= Y_1 + D \cos\alpha \end This is typically used in
triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...
and azimuth identification (AzID), especially in
radar Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, w ...
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 celestial equator is the great circle of the imaginary celestial sphere on the same plane as the equator of Earth. This plane of reference bases the equatorial coordinate system. In other words, the celestial equator is an abstract projectio ...
, the angles are called
right ascension Right ascension (abbreviated RA; symbol ) is the angular distance of a particular point measured eastward along the celestial equator from the Sun at the March equinox to the (hour circle of the) point in question above the earth. When paired w ...
if referenced to the Vernal Equinox, or hour angle if referenced to the
celestial meridian In astronomy, the meridian is the great circle passing through the celestial poles, as well as the zenith and nadir of an observer's location. Consequently, it contains also the true north, north and south points on the horizon, and it is perpe ...
.


Polar coordinate

In mathematics, the azimuth angle of a point in
cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
or
spherical coordinates 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'' measu ...
is the anticlockwise
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
between the positive ''x''-axis and the projection of the
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
onto the ''xy''-
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * Planes (gen ...
. The angle is the same as an angle in
polar coordinates 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 (analogous to the or ...
of the component of the vector in the ''xy''-plane and is normally measured in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s rather than degrees. As well as measuring the angle differently, in mathematical applications
theta Theta (, ; uppercase: Θ or ; lowercase: θ or ; grc, ''thē̂ta'' ; Modern: ''thī́ta'' ) is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth . In the system of Greek numerals, it has a value of 9. Gr ...
, ''θ'', is very often used to represent the azimuth rather than the representation of symbol
phi Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicele ...
''φ''.


Other uses

For magnetic tape drives, ''azimuth'' refers to the angle between the tape head(s) and tape. In
sound localization Sound localization is a listener's ability to identify the location or origin of a detected sound in direction and distance. The sound localization mechanisms of the mammalian auditory system have been extensively studied. The auditory system u ...
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 An azimuth thruster is a configuration of marine propellers placed in pods that can be rotated to any horizontal angle (azimuth), making a rudder unnecessary. These give ships better maneuverability than a fixed propeller and rudder system. Ty ...
in
shipbuilding Shipbuilding is the construction of ships and other floating vessels. It normally takes place in a specialized facility known as a shipyard. Shipbuilders, also called shipwrights, follow a specialized occupation that traces its roots to befor ...
is a
propeller A propeller (colloquially often called a screw if on a ship or an airscrew if on an aircraft) is a device with a rotating hub and radiating blades that are set at a pitch to form a helical spiral which, when rotated, exerts linear thrust upon ...
that can be rotated horizontally.


See also

*
Altitude (astronomy) The horizontal coordinate system is a celestial coordinate system that uses the observer's local horizon as the fundamental plane to define two angles: altitude and azimuth. Therefore, the horizontal coordinate system is sometimes called as the ...
*
Azimuthal quantum number The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe t ...
*
Azimuthal equidistant projection 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 azimut ...
*
Azimuth recording Azimuth recording is the use of a variation in angle between two recording heads that are recording data so close together on magnetic tape that crosstalk would otherwise likely occur. Normally, the head is perpendicular to the movement of the tape ...
*
Bearing (navigation) In navigation, bearing or azimuth is the horizontal angle between the direction of an object and north or another object. The angle value can be specified in various angular units, such as degrees, mils, or grad. More specifically: * Absolu ...
*
Clock position A clock position, or clock bearing, is the direction of an object observed from a vehicle, typically a vessel or an aircraft, relative to the orientation of the vehicle to the observer. The vehicle must be considered to have a front, a back, a ...
*
Course (navigation) In navigation, the course of a watercraft or aircraft is the cardinal direction in which the craft is to be steered. The course is to be distinguished from the ''heading'', which is the direction where the watercraft's bow or the aircraft's no ...
*
Inclination Orbital inclination measures the tilt of an object's orbit around a celestial body. It is expressed as the angle between a Plane of reference, reference plane and the orbital plane or Axis of rotation, axis of direction of the orbiting object ...
*
Longitude 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 l ...
*
Latitude 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 pol ...
*
Magnetic declination Magnetic declination, or magnetic variation, is the angle on the horizontal plane between magnetic north (the direction the north end of a magnetized compass needle points, corresponding to the direction of the Earth's magnetic field lines) and ...
*
Panning (camera) In cinematography and photography panning means swivelling a still or video camera horizontally from a fixed position. This motion is similar to the motion of a person when they turn their head on their neck from left to right. In the resulting ...
*
Relative bearing In navigation, bearing or azimuth is the horizontal angle between the direction of an object and north or another object. The angle value can be specified in various angular units, such as degrees, mils, or grad. More specifically: * Absolut ...
*
Sextant A sextant is a doubly reflecting navigation instrument that measures the angular distance between two visible objects. The primary use of a sextant is to measure the angle between an astronomical object and the horizon for the purposes of celes ...
*
Solar azimuth angle The solar azimuth angle is the azimuth (horizontal angle with respect to north) of the Sun's position. This horizontal coordinate defines the Sun's relative direction along the local horizon, whereas the solar zenith angle (or its complementary ...
*
Sound Localization Sound localization is a listener's ability to identify the location or origin of a detected sound in direction and distance. The sound localization mechanisms of the mammalian auditory system have been extensively studied. The auditory system u ...
*
Zenith The zenith (, ) is an imaginary point directly "above" a particular location, on the celestial sphere. "Above" means in the vertical direction (plumb line) opposite to the gravity direction at that location (nadir). The zenith is the "highest" ...


References


Further reading

* Rutstrum, Carl, ''The Wilderness Route Finder'', University of Minnesota Press (2000),


External links

* * {{Authority control Angle Navigation Surveying Horizontal coordinate system Geodesy