Azimuth
Encyclopedia
An azimuth is an angular measurement in a spherical coordinate system
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 inclination angle measured from a fixed zenith direction, and the azimuth angle of...

. The vector
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

from an observer (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...

) to a point of interest is projected
Projection (mathematics)
Generally speaking, in mathematics, a projection is a mapping of a set which is idempotent, which means that a projection is equal to its composition with itself. A projection may also refer to a mapping which has a left inverse. Bot notions are strongly related, as follows...

perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

ly onto a reference plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

; the angle between the projected vector and a reference vector on the reference plane is called the azimuth.

An example of an azimuth is the measurement of the position of a star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

in the sky. The star is the point of interest, the reference plane is the horizon
Horizon
The horizon is the apparent line that separates earth from sky, the line that divides all visible directions into two categories: those that intersect the Earth's surface, and those that do not. At many locations, the true horizon is obscured by trees, buildings, mountains, etc., and the resulting...

or the surface of the sea
Sea
A sea generally refers to a large body of salt water, but the term is used in other contexts as well. Most commonly, it means a large expanse of saline water connected with an ocean, and is commonly used as a synonym for ocean...

, and the reference vector points to the north
North
North is a noun, adjective, or adverb indicating direction or geography.North is one of the four cardinal directions or compass points. It is the opposite of south and is perpendicular to east and west.By convention, the top side of a map is north....

. The azimuth is the angle between the north point and the perpendicular projection of the star down onto the horizon.

Azimuth is usually measured in degrees
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...

(°). The concept is used in many practical applications including navigation
Navigation is the process of monitoring and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks...

, astronomy
Astrometry
Astrometry is the branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. The information obtained by astrometric measurements provides information on the kinematics and physical origin of our Solar System and our Galaxy, the Milky...

, engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, map
Map
A map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, regions, and themes....

ping, mining and artillery
Artillery
Originally applied to any group of infantry primarily armed with projectile weapons, artillery has over time become limited in meaning to refer only to those engines of war that operate by projection of munitions far beyond the range of effect of personal weapons...

.

In land navigation, azimuth is usually denoted as alpha
Alpha
Alpha is the first letter of the Greek alphabet. Alpha or ALPHA may also refer to:-Science:*Alpha , the highest ranking individuals in a community of social animals...

, , and defined as a horizontal angle measured clockwise from a north base line or 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...

. Azimuth has also been more generally defined as a horizontal angle measured clockwise from any fixed reference plane or easily established base direction line.

Today, the reference plane for an azimuth in a general navigational context is typically true north
True north
True north is the direction along the earth's surface towards the geographic North Pole.True geodetic north usually differs from magnetic north , and from grid north...

, measured as a 0° azimuth, though other angular units (grad
The gradian is a unit of plane angle, equivalent to of a turn. It is also known as gon, grad, or grade . One grad equals of a degree or of a radian...

, mil
Angular mil
An angular mil, also mil, is a unit of angle. All versions of the angular mil are approximately the same size as a trigonometric milliradian.-History:The milliradian was first identified in the mid nineteenth Century...

) can also be employed. In any event, the azimuth cannot exceed the highest number of units in a circle – for a 360° circle, this is 359 degrees, 59 arcminutes, 59 arcseconds (359° 59' 59").

For example, moving clockwise on a 360° degree circle, a point due east would have an azimuth of 90°, south 180°, and west 270°. However, there are exceptions: some navigation systems use geographic south as the reference plane.Any direction can potentially serve as the plane of reference, as long as it is clearly defined for everyone using that system.

### True north-based azimuths

From North
North 0° or 360° 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°

## Calculating Azimuth

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 is given by

A better approximation assumes the Earth is a slightly-squashed sphere (a 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 unmeasurably 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. the NGS site. (That site is simpler than it looks at first glance; its default is the GRS80/WGS84 spheroid, which is what most people want.) 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. If is the reciprocal of the flattening for the chosen spheroid (e.g. 298.257223563 for WGS84) then

If = 0 then

## Mapping

There are 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 as in the Philippine practice. However, any direction can serve as the plane of reference, as long as it is clearly defined for everyone using that system.

## Astronomy

Used in celestial navigation, an azimuth is the direction of a celestial body from the observer. In astronomy, an azimuth is sometimes referred to as a bearing
In marine navigation, a bearing is the direction one object is from another object, usually, the direction of an object from one's own vessel. In aircraft navigation, a bearing is the actual compass direction of the forward course of the aircraft...

. In modern astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

azimuth is nearly always measured from the north. 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
Hour angle
In astronomy and celestial navigation, the hour angle is one of the coordinates used in the equatorial coordinate system to give the position of a point on the celestial sphere....

of a star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

was zero. This assumes, however, that the star (upper) culminates
Culmination
In astronomy, the culmination of a planet, star, constellation, etc. is the altitude reached when the object transits over an observer's meridian....

in the south, which is only true for most stars in the Northern Hemisphere
Northern Hemisphere
The Northern Hemisphere is the half of a planet that is north of its equator—the word hemisphere literally means “half sphere”. It is also that half of the celestial sphere north of the celestial equator...

.

### Right Ascension

If instead of measuring from and along the horizon the angles are measured from and along the celestial equator
Celestial equator
The celestial equator is a great circle on the imaginary celestial sphere, in the same plane as the Earth's equator. In other words, it is a projection of the terrestrial equator out into space...

, the angles are called right ascension
Right ascension
Right ascension is the astronomical term for one of the two coordinates of a point on the celestial sphere when using the equatorial coordinate system. The other coordinate is the declination.-Explanation:...

if referenced to the Vernal Equinox, or hour angle if referenced to the celestial meridian.

### Horizontal coordinate

In the horizontal coordinate system
Horizontal coordinate system
The horizontal coordinate system is a celestial coordinate system that uses the observer's local horizon as the fundamental plane. This coordinate system divides the sky into the upper hemisphere where objects are visible, and the lower hemisphere where objects cannot be seen since the earth is in...

Celestial navigation, also known as astronavigation, is a position fixing technique that has evolved over several thousand years to help sailors cross oceans without having to rely on estimated calculations, or dead reckoning, to know their position...

and satellite dish
Satellite dish
A satellite dish is a dish-shaped type of parabolic antenna designed to receive microwaves from communications satellites, which transmit data transmissions or broadcasts, such as satellite television.-Principle of operation:...

installation, azimuth is one of the two coordinates
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...

. The other is altitude, sometimes called elevation above the horizon. See also: Sat finder
Sat finder
A Sat finder is a satellite signal meter used to accurately point satellite dishes at communications satellites in geostationary orbit.-External links:****...

.

### Polar coordinate

In three-dimensional polar coordinate systems, including cylindrical coordinates
Cylindrical coordinate system
A cylindrical coordinate system is a three-dimensional coordinate systemthat 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...

and spherical coordinates
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 inclination angle measured from a fixed zenith direction, and the azimuth angle of...

, the azimuth of a point is the angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

between the positive x-axis and the projection of the vector onto the xy-plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

(the component of the vector in the xy-plane). In cylindrical coordinates, theta
Theta
Theta is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth...

, , is almost universally used to represent the azimuth in mathematical applications, whereas physical applications may denote the azimuth using the symbol phi
Phi (letter)
Phi , pronounced or sometimes in English, and in modern Greek, is the 21st letter of the Greek alphabet. In modern Greek, it represents , a voiceless labiodental fricative. In Ancient Greek it represented , an aspirated voiceless bilabial plosive...

, . Although there are several conventions in spherical coordinates, the azimuth is usually denoted by either theta, , or phi, .

## Other uses of the term

The term azimuth is also used in context with military artillery
Artillery
Originally applied to any group of infantry primarily armed with projectile weapons, artillery has over time become limited in meaning to refer only to those engines of war that operate by projection of munitions far beyond the range of effect of personal weapons...

coordination. In artillery laying, an azimuth is defined as the direction of fire.

An azimuth in aerial navigation is defined as the direction of flight, as taken from the location of the aircraft.

In mining operations, an azimuth or meridian angle is any angle measured clockwise from any meridian or horizontal plane of reference.

For magnetic tape drives
Tape drive
A tape drive is a data storage device that reads and performs digital recording, writes data on a magnetic tape. Magnetic tape data storage is typically used for offline, archival data storage. Tape media generally has a favorable unit cost and long archival stability.A tape drive provides...

, 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
Azimuth thruster
An azimuth thruster is a configuration of ship propellers placed in pods that can be rotated in any horizontal direction, making a rudder unnecessary...

in shipbuilding
Shipbuilding
Shipbuilding is the construction of ships and 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 before recorded history.Shipbuilding and ship repairs, both...

is a propeller
Propeller
A propeller is a type of fan that transmits power by converting rotational motion into thrust. A pressure difference is produced between the forward and rear surfaces of the airfoil-shaped blade, and a fluid is accelerated behind the blade. Propeller dynamics can be modeled by both Bernoulli's...

that can be rotated horizontally.

• Altitude (astronomy)
• Azimuthal quantum number
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...

In marine navigation, a bearing is the direction one object is from another object, usually, the direction of an object from one's own vessel. In aircraft navigation, a bearing is the actual compass direction of the forward course of the aircraft...

In navigation, a vehicle's course is the angle that the intended path of the vehicle makes with a fixed reference object . Typically course is measured in degrees from 0° clockwise to 360° in compass convention . Course is customarily expressed in three digits, using preliminary zeros if needed,...

• Inclination
Inclination
Inclination in general is the angle between a reference plane and another plane or axis of direction.-Orbits:The inclination is one of the six orbital parameters describing the shape and orientation of a celestial orbit...

• 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 ....

• Magnetic declination
Magnetic declination
Magnetic declination is the angle between magnetic north and true north. The declination is positive when the magnetic north is east of true north. The term magnetic variation is a synonym, and is more often used in navigation...

• Panning (camera)
Panning (camera)
In photography, panning refers to the horizontal movement or rotation of a still or video camera, or the scanning of a subject horizontally on video or a display device...

• Solar azimuth angle
Solar azimuth angle
The solar azimuth angle is the azimuth angle of the sun. It is most often defined as the angle from due north in a clockwise direction.It can be calculated in various way. In different times, it has been explained in different ways. It can be calculated, to a good approximation, using the following...

• Sound Localization
Sound localization
Sound localization refers to a listener's ability to identify the location or origin of a detected sound in direction and distance. It may also refer to the methods in acoustical engineering to simulate the placement of an auditory cue in a virtual 3D space .The sound localization mechanisms of the...

• Zenith
Zenith
The zenith is an imaginary point directly "above" a particular location, on the imaginary celestial sphere. "Above" means in the vertical direction opposite to the apparent gravitational force at that location. The opposite direction, i.e...