|
|
| (673 intermediate revisions by more than 100 users not shown) |
| Line 1: |
Line 1: |
| {{About||the band|Azimuth (band)}}
| | This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users. |
|
| |
|
| [[Image:Azimuth-Altitude schematic.svg|right|350px|thumb|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.]] | | If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]] |
| | * Only registered users will be able to execute this rendering mode. |
| | * Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere. |
|
| |
|
| An '''azimuth''' ({{IPAc-en|audio=en-us-azimuth.ogg|ˈ|æ|z|ɪ|m|ə|θ}}; {{etymology|ara|''السمت'' as‑samt|a way, a part, or quarter}}<ref>{{cite book|title=Arts and sciences: or, Fourth division of "The English encyclopedia", Volume 1|author=Charles Knight|publisher=Bradbury, Evans & Co.|page=772}}</ref>) is an [[Angle#Measuring_angles|angular measurement]] in a [[spherical coordinate system]]. The [[vector space|vector]] from an observer ([[Origin (mathematics)|origin]]) to a point of interest is [[Projection (mathematics)|projected]] [[perpendicular]]ly onto a reference [[plane (mathematics)|plane]]; the angle between the projected vector and a reference vector on the reference plane is called the azimuth.
| | Registered users will be able to choose between the following three rendering modes: |
|
| |
|
| 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 point and the perpendicular projection of the star down onto the horizon.<ref>http://dictionary.reference.com/browse/azimuth</ref>
| | '''MathML''' |
| | :<math forcemathmode="mathml">E=mc^2</math> |
|
| |
|
| Azimuth is usually measured in [[degree (angle)|degrees]] (°). The concept is used in many practical applications including [[navigation]], [[astrometry|astronomy]], [[engineering]], [[map]]ping, mining and [[artillery]].
| | <!--'''PNG''' (currently default in production) |
| | :<math forcemathmode="png">E=mc^2</math> |
|
| |
|
| ==Navigation==
| | '''source''' |
| In land navigation, azimuth is usually denoted as ''[[alpha]]'', <math>\alpha</math>, and defined as a horizontal angle measured [[clockwise and counterclockwise|clockwise]] from a north base line or ''[[meridian (geography)|meridian]]''.<ref>U.S. Army, ''Map Reading and Land Navigation'', FM 21-26, Headquarters, Dept. of the Army, Washington, D.C. (7 May 1993), ch. 6, p. 2</ref><ref>U.S. Army, ''Map Reading and Land Navigation'', FM 21-26, Headquarters, Dept. of the Army, Washington, D.C. (28 March 1956), ch. 3, p. 63</ref> ''Azimuth'' has also been more generally defined as a horizontal angle measured clockwise from any fixed reference plane or easily established base direction line.<ref>U.S. Army, ch. 6 p. 2</ref><ref>U.S. Army, ''Advanced Map and Aerial Photograph Reading'', Headquarters, War Department, Washington, D.C. (17 September 1941), pp. 24-25</ref><ref>U.S. Army, ''Advanced Map and Aerial Photograph Reading'', Headquarters, War Department, Washington, D.C. (23 December 1944), p. 15</ref>
| | :<math forcemathmode="source">E=mc^2</math> --> |
|
| |
|
| 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 (angle)|grad]], [[Angular mil|mil]]) can also be employed.
| | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
|
| |
|
| 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.
| | ==Demos== |
|
| |
|
| ===True north-based azimuths===
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
| {| class="wikitable" style="margin:auto;"
| |
| |-----
| |
| ! colspan="4" | From North
| |
| |-----
| |
| | North || align="center" | 0° or 360°
| |
| || South || align="center" | 180°
| |
| |-----
| |
| | North-Northeast || align="center" | 22.5°
| |
| || South-Southwest || align="center" | 202.5°
| |
| |-----
| |
| | Northeast || align="center" | 45°
| |
| || Southwest || align="center" | 225°
| |
| |-----
| |
| | East-Northeast || align="center" | 67.5°
| |
| || West-Southwest || align="center" | 247.5°
| |
| |-----
| |
| | East || align="center" | 90°
| |
| || West || align="center" | 270°
| |
| |-----
| |
| | East-Southeast || align="center" | 112.5°
| |
| || West-Northwest || align="center" | 292.5°
| |
| |-----
| |
| | Southeast || align="center" | 135°
| |
| || Northwest || align="center" | 315°
| |
| |-----
| |
| | South-Southeast || align="center" | 157.5°
| |
| || North-Northwest || align="center" | 337.5°
| |
| |}
| |
|
| |
|
| ==Calculating Azimuth==
| |
|
| |
|
| We are standing at latitude <math> \phi_1 </math>, longitude zero; we want to find the azimuth from our viewpoint to Point 2 at latitude <math> \phi_2 </math>, longitude L (positive eastward). We can get a fair approximation by assuming the Earth is a sphere, in which case the azimuth <math> \alpha </math> is given by
| | * accessibility: |
| | ** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)] |
| | ** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]]. |
| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
|
| |
|
| :<math>\tan \alpha
| | ==Test pages == |
| = \frac{\sin L}{(\cos \phi_1)(\tan \phi_2)- (\sin\phi_1)(\cos L)}</math> | |
|
| |
|
| 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.
| | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
| | *[[Displaystyle]] |
| | *[[MathAxisAlignment]] |
| | *[[Styling]] |
| | *[[Linebreaking]] |
| | *[[Unique Ids]] |
| | *[[Help:Formula]] |
|
| |
|
| Various websites will calculate geodetic azimuth—e.g. [http://www.ngs.noaa.gov/cgi-bin/Inv_Fwd/inverse2.prl 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 [[Geographical distance#Ellipsoidal-surface formulae|distance article]].
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| | | *[[Url2Image|Url2Image (private Wikis only)]] |
| Normal-section azimuth is simpler to calculate; Bomford says Cunningham's formula is exact for any distance. If <math> r</math> is the reciprocal of the flattening for the chosen spheroid (e.g. 298.257223563 for WGS84) then
| | ==Bug reporting== |
| | | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
| <math> e^2 \quad = \quad \cfrac {2r - 1}{r^2} </math>
| |
| | |
| <math> (1 - e^2) \quad = \quad \left ( \frac {r - 1}{r} \right )^2 </math>
| |
| | |
| <math> \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}} </math>
| |
| | |
| <math> \tan \alpha \quad = \quad \frac {\sin L}{(\Lambda - \cos L) \sin \phi_1 }</math><br><br>
| |
| | |
| If <math>\phi_1</math> = 0 then
| |
| | |
| <math> \tan \alpha \quad = \quad \frac {\sin L}{(1 - e^2) \tan \phi_2}</math>
| |
| | |
| ==Mapping==
| |
| [[Image:Brunton.JPG|thumb|A standard Brunton Geo [[compass]], used commonly by geologists and surveyors to measure azimuth]]
| |
| There are a wide variety of [[Map projection#Azimuthal .28projections onto a plane.29|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.<ref>Rutstrum, Carl, The Wilderness Route Finder, University of Minnesota Press (2000), ISBN 0-8166-3661-3, p. 194</ref> In astronomy, an ''azimuth'' is sometimes referred to as a [[bearing (navigation)|bearing]]. In modern [[astronomy]] 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]] of a [[star]] was zero. This assumes, however, that the star [[culmination|(upper) culminates]] in the south, which is only true for most stars in the [[Northern Hemisphere]].
| |
| | |
| ==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 [[coordinate system|coordinates]]. The other is [[Altitude (astronomy)|altitude]], sometimes called elevation above the horizon. See also: [[Sat finder]].
| |
| | |
| ===Polar coordinate===
| |
| In mathematics the azimuth angle of a point in [[cylindrical coordinate system|cylindrical coordinates]] or [[spherical coordinate system|spherical coordinates]] is the anticlockwise [[angle]] between the positive x-axis and the projection of the [[Vector (geometry)|vector]] onto the xy-[[plane (mathematics)|plane]]. The angle is the same was as an angle in [[polar coordinates]] of the component of the vector in the xy-plane and is normally measured in [[radian]]s rather than degrees. As well as measuring the angle differently, in mathematical applications [[theta]], <math>\theta</math>, is very often used to represent the azimuth rather than the symbol [[phi (letter)|phi]] <math>\phi</math>.
| |
| | |
| ==Other uses of the term== | |
| The term ''azimuth'' is also used in context with military [[artillery]] 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 [[tape drive|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==
| |
| {{Portal|Nautical}}
| |
| * [[Altitude (astronomy)]]
| |
| * [[Azimuthal quantum number]]
| |
| * [[Bearing (navigation)]]
| |
| * [[Course (navigation)]]
| |
| * [[Inclination]]
| |
| * [[Longitude]]
| |
| * [[Magnetic declination]]
| |
| * [[Panning (camera)]]
| |
| * [[Solar azimuth angle]]
| |
| * [[Sound localization|Sound Localization]]
| |
| * [[Zenith]]
| |
| | |
| ==Notes==
| |
| {{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 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==
| |
| {{Wiktionary|azimuth}}
| |
| *{{Cite EB1911|Azimuth}}
| |
| *{{Cite Collier's|Azimuth|year=1921}}
| |
| | |
| [[Category:Navigation]]
| |
| | |
| <!-- interwiki -->
| |
| | |
| [[af:Asimut]]
| |
| [[als:Azimut]]
| |
| [[ar:سمت]]
| |
| [[ast:Acimut]]
| |
| [[ca:Azimut]]
| |
| [[cs:Azimut]]
| |
| [[da:Azimut (astronomi)]]
| |
| [[de:Azimut]]
| |
| [[et:Asimuut]]
| |
| [[el:Αζιμούθιο]]
| |
| [[es:Acimut]]
| |
| [[eo:Azimuta angulo]]
| |
| [[ext:Azimu]]
| |
| [[eu:Azimut]]
| |
| [[fa:سمت (ستارهشناسی)]]
| |
| [[fr:Azimut]]
| |
| [[gl:Acimut]]
| |
| [[ko:방위각]]
| |
| [[hi:दिगंश]]
| |
| [[hy:Ազիմուտ]]
| |
| [[hr:Azimut]]
| |
| [[io:Azimuto]]
| |
| [[id:Azimut]]
| |
| [[is:Áttarhorn]]
| |
| [[it:Azimut]]
| |
| [[he:אזימוט]]
| |
| [[csb:Azymùt]]
| |
| [[kk:Азимут]]
| |
| [[lv:Azimuts]]
| |
| [[lb:Azimut]]
| |
| [[lt:Azimutas]]
| |
| [[hu:Azimut]]
| |
| [[mk:Азимут]]
| |
| [[mn:Азимут]]
| |
| [[nl:Azimut]]
| |
| [[ja:方位角]]
| |
| [[no:Asimut]]
| |
| [[nn:Asimut]]
| |
| [[pl:Azymut]]
| |
| [[pt:Azimute]]
| |
| [[ru:Азимут (геодезия)]]
| |
| [[sah:Аазимут]]
| |
| [[sq:Azimuti]]
| |
| [[sk:Azimut (astronómia)]]
| |
| [[sl:Azimut]]
| |
| [[sr:Азимут]]
| |
| [[sh:Azimut]]
| |
| [[fi:Atsimuutti]]
| |
| [[sv:Azimut]]
| |
| [[ta:திசைவில்]]
| |
| [[tt:Азимут (геодезия)]]
| |
| [[th:มุมทิศ]]
| |
| [[tr:Azimut]]
| |
| [[uk:Азимут]]
| |
| [[ur:سمت]]
| |
| [[zh:方位角]]
| |
