Azimuth

Last updated
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 zenith. Azimuth-Altitude schematic.svg
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 zenith.

An azimuth ( /ˈæzəməθ/ ( Loudspeaker.svg listen ); from Arabic : اَلسُّمُوت, romanized: as-sumūt, lit. 'the directions') [1] 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.

Contents

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 with a 5 km 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. [2]

Azimuth is usually measured in degrees (°). The concept is used in navigation, astronomy, engineering, mapping, mining, and ballistics.

Etymology

The word azimuth is used in all European languages today. It originates from medieval Arabic السموت (al-sumūt, pronounced as-sumūt), meaning "the directions" (plural of Arabic السمت 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. Its first recorded use in English is in the 1390s in Geoffrey Chaucer's Treatise on the Astrolabe . 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 astronomía commissioned by King Alfonso X of Castile. [3]

In navigation

Azimuth Marker, Mount Allen (Sandstone Peak), Southern California, US. True North Mount Allen.fw.png
Azimuth Marker, Mount Allen (Sandstone Peak), Southern California, US.

In land navigation, azimuth is usually denoted alpha, α, and defined as a horizontal angle measured clockwise from a north base line or meridian . [4] [5] Azimuth has also been more generally defined as a horizontal angle measured clockwise from any fixed reference plane or easily established base direction line. [6] [7] [8]

Today, the reference plane for an azimuth is typically true north, measured as a 0° azimuth, though other angular units (grad, mil) can be used. Moving clockwise on a 360 degree circle, east has azimuth 90°, south 180°, and west 270°. There are exceptions: some navigation systems use south as the reference vector. Any direction can be the reference vector, as long as it is clearly defined.

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 described as "(from) south, (turn) thirty degrees (toward the) east" (the words in brackets are usually omitted), abbreviated "S30°E", which is the bearing 30 degrees in the eastward direction from south, i.e. the bearing 150 degrees clockwise from north. The reference direction, stated first, is always north or south, and the turning 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, eastern side
DirectionAzimuth
North
North-northeast22.5°
Northeast45°
East-northeast67.5°
East90°
East-southeast112.5°
Southeast135°
South-southeast157.5°
From north, western side
DirectionAzimuth
South180°
South-southwest202.5°
Southwest225°
West-southwest247.5°
West270°
West-northwest292.5°
Northwest315°
North-northwest337.5°

In geodesy

The azimuth between Cape Town and Melbourne along the geodesic (the shortest route) changes from 141deg to 42deg. Azimuthal orthographic projection and Miller cylindrical projection. Bearing and azimuth along the geodesic.png
The azimuth between Cape Town and Melbourne along the geodesic (the shortest route) changes from 141° to 42°. Azimuthal orthographic projection and Miller cylindrical projection.

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 α is given by

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 (or geodesic azimuth) is the angle between north and the ellipsoidal geodesic (the shortest path on the surface of the spheroid from our viewpoint to Point 2). The difference is usually negligible: less than 0.03 arc second for distances less than 100 km. [9]

Normal-section azimuth can be calculated as follows:[ citation needed ]

where f is the flattening and e the eccentricity for the chosen spheroid (e.g., 1298.257223563 for WGS84). If φ1 = 0 then

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).[ citation needed ]

In cartography

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

The cartographical azimuth or grid azimuth (in decimal degrees) can be calculated when the coordinates of 2 points are known in a flat plane (cartographical coordinates):

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:

Note the swapped in contrast to the normal atan2 input order.

The opposite problem occurs when the coordinates (X1, Y1) of one point, the distance D, and the azimuth α to another point (X2, Y2) are known, one can calculate its coordinates:

This is typically used in triangulation and azimuth identification (AzID), especially in radar applications.

Map projections

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.

Comparison azimuthal projections.svg
Comparison of some azimuthal projections centred on 90° N at the same scale, ordered by projection altitude in Earth radii. (click for detail)

In astronomy

In the horizontal coordinate system, used in celestial navigation, azimuth is one of the two coordinates. [10] The other is altitude , sometimes called elevation above the horizon. It is also used for satellite dish installation (see also: sat finder). In modern astronomy azimuth is nearly always measured from the north.

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.

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, θ, is very often used to represent the azimuth rather than the representation of symbol phi φ.

Other uses

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

Notes

  1. The singular form of the noun is Arabic : السَّمْت, romanized: as-samt, lit. 'the direction'.
  2. "azimuth". Dictionary.com Unabridged (Online). n.d.
  3. "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 Archived January 2, 2014, at the Wayback Machine . In Arabic the written al-sumūt is always pronounced as-sumūt (see pronunciation of "al-" in Arabic).
  4. 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
  5. 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
  6. U.S. Army, ch. 6 p. 2
  7. U.S. Army, Advanced Map and Aerial Photograph Reading, Headquarters, War Department, Washington, D.C. (17 September 1941), pp. 24–25
  8. U.S. Army, Advanced Map and Aerial Photograph Reading, Headquarters, War Department, Washington, D.C. (23 December 1944), p. 15
  9. Torge & Müller (2012) Geodesy, De Gruyter, eq.6.70, p.248
  10. Rutstrum, Carl, The Wilderness Route Finder, University of Minnesota Press (2000), ISBN   0-8166-3661-3, p. 194

Related Research Articles

Mercator projection Cylindrical conformal map projection

The Mercator projection is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because it is unique in representing north as up and south as down everywhere while preserving local directions and shapes. The map is thereby conformal. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation is very small near the equator but accelerates with increasing latitude to become infinite at the poles. As a result, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator, such as Central Africa.

Polar coordinate system Coordinates determined by distance and angle

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. Angles in polar notation are generally expressed in either degrees or radians.

Spherical coordinate system 3-dimensional 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 polar angle measured from a fixed zenith direction, and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. An elliptically polarized wave may be resolved into two linearly polarized waves in phase quadrature, with their polarization planes at right angles to each other. Since the electric field can rotate clockwise or counterclockwise as it propagates, elliptically polarized waves exhibit chirality.

Astronomical coordinate systems System for specifying positions of celestial objects

Astronomical coordinate systems are organized arrangements for specifying positions of satellites, planets, stars, galaxies, and other celestial objects relative to physical reference points available to a situated observer. Coordinate systems in astronomy can specify an object's position in three-dimensional space or plot merely its direction on a celestial sphere, if the object's distance is unknown or trivial.

In physics, angular velocity or rotational velocity, also known as angular frequency vector, is a pseudovector representation of how fast the angular position or orientation of an object changes with time. The magnitude of the pseudovector represents the angular speed, the rate at which the object rotates or revolves, and its direction is normal to the instantaneous plane of rotation or angular displacement. The orientation of angular velocity is conventionally specified by the right-hand rule.

Unit vector Vector of length one

In mathematics, a unit vector in a normed vector space is a vector of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in .

Cylindrical coordinate system 3-dimensional coordinate system

A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis (axis L in the image opposite), the direction from the axis relative to a chosen reference direction (axis A), and the distance from a chosen reference plane perpendicular to the axis (plane containing the purple section). The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point.

Rhumb line Arc crossing all meridians of longitude at the same angle

In navigation, a rhumb line, rhumb, or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true north.

Dihedral angle Angle between two planes in space

A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the union of a line and two half-planes that have this line as a common edge. In higher dimensions, a dihedral angle represents the angle between two hyperplanes. The planes of a flying machine are said to be at positive dihedral angle when both starboard and port main planes are upwardly inclined to the lateral axis. When downwardly inclined they are said to be at a negative dihedral angle.

Euler angles Description of the orientation of a rigid body

The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.

Orthographic map projection Azimuthal perspective map projection

Orthographic projection in cartography has been used since antiquity. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective projection in which the sphere is projected onto a tangent plane or secant plane. The point of perspective for the orthographic projection is at infinite distance. It depicts a hemisphere of the globe as it appears from outer space, where the horizon is a great circle. The shapes and areas are distorted, particularly near the edges.

The solar zenith angle is the angle between the sun’s rays and the vertical direction. It is closely related to the solar altitude angle, which is the angle between the sun’s rays and a horizontal plane. Since these two angles are complementary, the cosine of either one of them equals the sine of the other. They can both be calculated with the same formula, using results from spherical trigonometry. At solar noon, the zenith angle is at a minimum and is equal to latitude minus solar declination angle. This is the basis by which ancient mariners navigated the oceans.

The solar azimuth angle is the azimuth angle of the Sun's position. This horizontal coordinate defines the Sun's relative direction along the local horizon, whereas the solar zenith angle defines the Sun's apparent altitude.

Azimuthal equidistant projection Azimuthal equidistant map projection

The azimuthal equidistant projection is an azimuthal map projection. It has the useful properties that all points on the map are at proportionally correct distances from the center point, and that all points on the map are at the correct azimuth (direction) from the center point. A useful application for this type of projection is a polar projection which shows all meridians as straight, with distances from the pole represented correctly. The flag of the United Nations contains an example of a polar azimuthal equidistant projection.

Scale (map) Ratio of distance on a map to the corresponding distance on the ground

The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways.

Position angle Convention for measuring angles on the sky

In astronomy, position angle is the convention for measuring angles on the sky. The International Astronomical Union defines it as the angle measured relative to the north celestial pole (NCP), turning positive into the direction of the right ascension. In the standard (non-flipped) images, this is a counterclockwise measure relative to the axis into the direction of positive declination.

There are several equivalent ways for defining trigonometric functions, and the proof of the trigonometric identities between them depend on the chosen definition. The oldest and somehow the most elementary definition is based on the geometry of right triangles. The proofs given in this article use this definition, and thus apply to non-negative angles not greater than a right angle. For greater and negative angles, see Trigonometric functions.

In mathematics and statistics, a circular mean or angular mean is a mean designed for angles and similar cyclic quantities, such as daytimes, and fractional parts of real numbers. This is necessary since most of the usual means may not be appropriate on angle-like quantities. For example, the arithmetic mean of 0° and 360° is 180°, which is misleading because 360° equals 0° modulo a full cycle. As another example, the "average time" between 11 PM and 1 AM is either midnight or noon, depending on whether the two times are part of a single night or part of a single calendar day. The circular mean is one of the simplest examples of circular statistics and of statistics of non-Euclidean spaces.

In spherical astronomy, the parallactic angle is the angle between the great circle through a celestial object and the zenith, and the hour circle of the object. It is usually denoted q. In the triangle zenith—object—celestial pole, the parallactic angle will be the position angle of the zenith at the celestial object. Despite its name, this angle is unrelated with parallax. The parallactic angle is zero or 180° when the object crosses the meridian.

References