An azimuth (/ˈæzɪməθ/ ( listen)) (from the pl. form of
the Arabic noun "السَّمْت" 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.
An example of azimuth is the angular direction of a star in the sky.
The star is the point of interest, the reference plane is the local
horizontal area (e.g. a circular area 5 km in radius around an
observer at sea level), and the reference vector points north. The
azimuth is the angle between the north vector and the star's vector on
the horizontal plane.[1]
Contents 1 Navigation 1.1 True north-based azimuths 2 Cartographical azimuth 2.1 Calculating coordinates 3 Calculating azimuth 4 Mapping 5 Astronomy 6 Other systems 6.1 Right ascension 6.2 Horizontal coordinate 6.3 Polar coordinate 7 Other uses of the word 8 Etymology of the word 9 See also 10 Notes 11 References 12 External links Navigation[edit]
In land navigation, azimuth is usually denoted alpha, α, and defined
as a horizontal angle measured clockwise from a north base line or
meridian.[2][3]
From north North 0° 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° Cartographical azimuth[edit] The cartographical azimuth (in decimal degrees) can be calculated when the coordinates of 2 points are known in a flat plane (cartographical coordinates): α = 180 π atan2 ( X 2 − X 1 , Y 2 − Y 1 ) displaystyle 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 360°. The formula in radians would be slightly easier: α = atan2 ( X 2 − X 1 , Y 2 − Y 1 ) displaystyle 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[edit] When the coordinates (X1, Y1) of one point, the distance L, and the azimuth α to another point (X2, Y2) are known, one can calculate its coordinates: X 2 = X 1 + L sin α Y 2 = Y 1 + L cos α displaystyle begin aligned X_ 2 &=X_ 1 +Lsin alpha \Y_ 2 &=Y_ 1 +Lcos alpha end aligned This is typically used in triangulation. Calculating azimuth[edit] The azimuth between
We are standing at latitude φ 1 displaystyle varphi _ 1 , longitude zero; we want to find the azimuth from our viewpoint to Point 2 at latitude φ 2 displaystyle 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 α = sin L cos φ 1 tan φ 2 − sin φ 1 cos L displaystyle tan alpha = frac sin L cos varphi _ 1 tan varphi _ 2 -sin varphi _ 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]. If f is the flattening for the chosen spheroid (e.g., 1⁄7002298257223563000♠298.257223563 for WGS84) then e 2 = f ( 2 − f ) 1 − e 2 = ( 1 − f ) 2 Λ = ( 1 − e 2 ) tan φ 2 tan φ 1 + e 2 1 + ( 1 − e 2 ) ( tan φ 2 ) 2 1 + ( 1 − e 2 ) ( tan φ 1 ) 2 tan α = sin L ( Λ − cos L ) sin φ 1 displaystyle begin aligned e^ 2 &=f(2-f)\1-e^ 2 &=(1-f)^ 2 \Lambda &=left(1-e^ 2 right) frac tan varphi _ 2 tan varphi _ 1 +e^ 2 sqrt cfrac 1+left(1-e^ 2 right)left(tan varphi _ 2 right)^ 2 1+left(1-e^ 2 right)left(tan varphi _ 1 right)^ 2 \tan alpha &= frac sin L (Lambda -cos L)sin varphi _ 1 end aligned If φ1 = 0 then tan α = sin L ( 1 − e 2 ) tan φ 2 displaystyle tan alpha = frac sin L left(1-e^ 2 right)tan varphi _ 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 φ2 with declination and longitude difference with hour angle, and change the sign (since the hour angle is positive westward instead of east). Mapping[edit] 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. A standard Brunton Geo compass, commonly used by geologists and surveyors to measure azimuth Comparison of some azimuthal projections centred on 90° N at the same scale, ordered by projection altitude in Earth radii. (click for detail) Astronomy[edit]
Used in celestial navigation, an azimuth is the direction of a
celestial body from the observer.[7] 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[edit]
Right ascension[edit]
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[edit]
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[edit]
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, θ, is very often
used to represent the azimuth rather than the representation of symbol
phi φ.
Other uses of the word[edit]
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[edit]
The word azimuth is in all European languages today. It originates
from medieval Arabic al-sumūt, pronounced as-sumūt 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
Nautical portal Altitude (astronomy) Azimuthal quantum number Azimuthal equidistant projection Bearing (navigation) Course (navigation) Inclination Longitude Latitude Magnetic declination Panning (camera) Sextant Solar azimuth angle Sound Localization Zenith Notes[edit] ^ "Azimuth". Dictionary.com.
^ U.S. Army,
References[edit] Rutstrum, Carl, The Wilderness Route Finder, University of Minnesota
Press (2000), ISBN 0-8166-3661-3
U.S. Army, Advanced
External links[edit] Look up azimuth in Wiktionary, the free dictionary. "Azimuth". Encyclopædia Britannica (11th ed.). 1911. "Azimuth". Collier's New Encyclo |