Azimuth

For the band, see Azimuth (band).

The azimuth is the angle formed between a reference direction (North) and a line from the observer to a point of interest projected on the same plane as the reference direction.

An azimuth (ⁱ/ˈæzɪməθ/; from Arabic السمت as‑samt, meaning "a way, a part, or quarter") 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.

An example of an azimuth is the measurement of the position of a star in the sky. The star is the point of interest, the reference plane is the horizon or the surface of the sea, and the reference vector points to the north. The azimuth is the angle between the north vector and the perpendicular projection of the star down onto the horizon.

Azimuth is usually measured in degrees (°). The concept is used in many practical applications including navigation, astronomy, engineering, mapping, mining and artillery.

Navigation

In land navigation, azimuth is usually denoted as alpha, $\alpha$, and defined as a horizontal angle measured clockwise from a north base line or meridian. 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, measured as a 0° azimuth, though other angular units (grad, mil) can also be employed.

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.

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 "south, thirty degrees east", abbreviated "S30°E", which is the bearing 30 degrees in the eastward direction from south, i.e. a bearing of 150 degrees clockwise from north. The reference direction, stated first, is always north or south, and the other 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.

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 $\phi_1$, longitude zero; we want to find the azimuth from our viewpoint to Point 2 at latitude $\phi_2$, longitude L (positive eastward). We can get a fair approximation by assuming the Earth is a sphere, in which case the azimuth $\alpha$ is given by

$\tan \alpha = \frac{\sin L}{(\cos \phi_1)(\tan \phi_2)- (\sin\phi_1)(\cos L)}$

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 conceptual difference is that a plumb-line measures a gradient in the gravitational potential associated with the Newtonian forces, whereas the shortest path to the surface is a purely geometric entity. 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. GeoScience Austriala 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. If $r$ is the reciprocal of the flattening for the chosen spheroid (e.g. 298.257223563 for WGS84) then

$e^2 \quad = \quad \cfrac {2r - 1}{r^2}$

$(1 - e^2) \quad = \quad \left ( \frac {r - 1}{r} \right )^2$

$\Lambda \quad = \quad (1 - e^2) \frac { \tan \phi_2}{ \tan \phi_1} \quad + \quad e^2 \sqrt{ \cfrac {1 + (1 - e^2)(\tan \phi_2)^2}{1 + (1 - e^2)(\tan \phi_1)^2}}$

$\tan \alpha \quad = \quad \frac {\sin L}{(\Lambda - \cos L) \sin \phi_1 }$

If $\phi_1$ = 0 then

$\tan \alpha \quad = \quad \frac {\sin L}{(1 - e^2) \tan \phi_2}$

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 $\phi_2$ with declination and longitude difference with hour angle, and change the sign (since hour angle is positive westward instead of east).

Mapping

A standard Brunton Geo compass, used commonly by geologists and surveyors to measure azimuth

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. 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 modern astronomy azimuth is nearly always measured from the north. (The article on coordinate systems, for example, uses a convention measuring form 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.

Other systems

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.

Horizontal coordinate

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.

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. 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, $\theta$, is very often used to represent the azimuth rather than the symbol phi $\phi$.

Other uses of the term

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.

See also

• Altitude (astronomy)
• Azimuthal quantum number
• Bearing (navigation)
• Course (navigation)
• Inclination
• Longitude
• Magnetic declination
• Panning (camera)
• Solar azimuth angle
• Sound Localization
• Zenith

References

• 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)

External links

•  Chisholm, Hugh, ed. (1911). "Azimuth". Encyclopædia Britannica (11th ed.). Cambridge University Press. 
•  "Azimuth". Collier's New Encyclopedia. 1921.