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### azimuth

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azimuth
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{{other uses}}File:Azimuth-Altitude schematic.svg|right|350px|thumb|The azimuth is the angle formed between a reference direction (in this example north) and a line from the observer to a point of interest projected on the same plane as the reference direction orthogonal to the zenithzenith{{Astrodynamics}}An azimuth ({{IPAc-en|audio=en-us-azimuth.ogg|Ëˆ|Ã¦|z|Éª|m|É™|Î¸}}; from Arabic Ø§ÙŽÙ„Ø³ÙÙ‘Ù…ÙÙˆØªâ€Ž as-sumÅ«t, â€œthe directionsâ€, the plural form of the Arabic noun Ø§Ù„Ø³ÙŽÙ‘Ù…Ù’Øª 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.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.WEB,weblink Azimuth, Dictionary.com, Azimuth is usually measured in degrees (Â°). The concept is used in navigation, astronomy, engineering, mapping, mining, and ballistics.

0Â° 180Â°
22.5Â° 202.5Â°
45Â° 225Â°
67.5Â° 247.5Â°
90Â° 270Â°
112.5Â° 292.5Â°
135Â° 315Â°
157.5Â° 337.5Â°

## Cartographical azimuth

The cartographical azimuth (in decimal degrees) can be calculated when the coordinates of 2 points are known in a flat plane (cartographical coordinates):
alpha = frac{180}{pi} operatorname{atan2}(X_2 - X_1, Y_2 - Y_1)
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 360Â°.The formula in radians would be slightly easier:
alpha = operatorname{atan2}(X_2 - X_1, Y_2 - Y_1)
Caveat: Most computer libraries (C/C++, Python, Java, ...) reverse the order of the atan2 parameters.

### Calculating coordinates

When the coordinates (X1, Y1) of one point, the distance L, and the azimuth Î± to another point (X2, Y2) are known, one can calculate its coordinates:
begin{align}
X_2 &= X_1 + L sinalpha
Y_2 &= Y_1 + L cosalpha
end{align}This is typically used in triangulation.

## Calculating azimuth

File:Bearing and azimuth along the geodesic.png|thumb|The azimuth between Cape Town and Melbourne along the geodesic (the shortest route) changes from 141Â° to 42Â°. Azimuthal orthographic projection and Miller cylindrical projectionMiller cylindrical projectionWe 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 Î± is given by
tanalpha = frac{sin L}{cosvarphi_1 tanvarphi_2 - sinvarphi_1 cos L}
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|reason=No mention of (Guy) Bomford or Cunningham anywhere else on this page|date=September 2015}} If f is the flattening, and e the eccentricity, for the chosen spheroid (e.g., {{frac|{{val|298.257223563}}}} for WGS84) then
begin{align}
e^2 &= f(2 - f)
1 - e^2 &= (1 - f)^2
Lambda &= left(1 - e^2right) frac{tanvarphi_2}{tanvarphi_1} + e^2
sqrt{frac{1 + left(1 - e^2right)left(tanvarphi_2right)^2}
{1 + left(1 - e^2right)left(tanvarphi_1right)^2}}
tanalpha &= frac{sin L}{(Lambda - cos L)sinvarphi_1}
end{align}If Ï†1 = 0 then
tanalpha = frac{sin L}{left(1 - e^2right)tanvarphi_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 Ï†2 with declination and longitude difference with hour angle, and change the sign (since the hour angle is positive westward instead of east).

## Mapping

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.File:Brunton.JPG|thumb|right|A standard Brunton Geo compasscompass{|align=left820px|}}{{clear}}

## Astronomy

Used in celestial navigation, an azimuth is the direction of a celestial body from the observer.Rutstrum, Carl, The Wilderness Route Finder, University of Minnesota Press (2000), {{ISBN|0-8166-3661-3}}, p. 194 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.

## 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, Î¸, is very often used to represent the azimuth rather than the representation of symbol phi Ï†.

## Other uses of the word

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.

## Etymology of the word

The word azimuth is in all European languages today. It originates from medieval Arabic al-sumÅ«t, pronounced as-sumÅ«t 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 astronomÃ­a commissioned by King Alfonso X of Castile."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 {{webarchive |url=https://web.archive.org/web/20140102020035weblink |date=January 2, 2014 }}. In Arabic the written al-sumÅ«t is always pronounced as-sumÅ«t (see pronunciation of "al-" in Arabic).

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{{Reflist}}

## 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 21â€“26, Headquarters, War Department, Washington, D.C. (17 September 1941)
• U.S. Army, Advanced Map and Aerial Photograph Reading, FM 21â€“26, Headquarters, War Department, Washington, D.C. (23 December 1944)
• U.S. Army, Map Reading and Land Navigation, FM 21â€“26, Headquarters, Dept. of the Army, Washington, D.C. (7 May 1993)

{{Wiktionary|azimuth}}
• EB1911, Azimuth, x,
• COLLIER'S, Azimuth, 1921, x,

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