Editing Spherical coordinate system (section)

== Applications ==
[[File:Spherical coordinate system.svg|thumb|upright=1.2|right|In the '''''mathematics convention''''': A globe showing a [[unit sphere]], with [[tuple]] coordinates of point {{mvar|P}} (red): its radial distance {{mvar|r}} (red, not labeled);  its azimuthal angle {{mvar|θ}} (not labeled);  and its polar angle of ''inclination'' {{mvar|φ}} (not labeled).  The radial distance upward along the [[zenith|zenith{{ndash}}axis]] from the point of origin to the surface of the sphere is assigned the value unity, or 1. + In this image, {{mvar|r}} appears to equal 4/6, or .67, (of unity); i.e., four of the six 'nested shells' to the surface. The azimuth angle {{mvar|θ}} appears to equal positive 90°, as rotated ''counterclockwise'' from the azimuth-reference x{{ndash}}axis;  and the inclination {{mvar|φ}} appears to equal 30°, as rotated from the zenith{{ndash}}axis. (Note the 'full' rotation, or inclination, from the zenith{{ndash}}axis to the y{{ndash}}axis is 90°).]]

Just as the two-dimensional [[Cartesian coordinate system]] is useful{{mdash}}has a wide set of applications{{mdash}}on a planar surface, a two-dimensional spherical coordinate system is useful on the surface of a sphere. For example, one sphere that is described in ''Cartesian coordinates''  with the equation {{math|''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> {{=}} ''c''<sup>2</sup>}} can be described in ''spherical coordinates'' by the simple equation {{math|''r'' {{=}} ''c''}}. (In this system{{mdash}}''shown here in the mathematics convention''{{mdash}}the sphere is adapted as a [[unit sphere]], where the radius is set to unity and then can generally be ignored, see graphic.)

This (unit sphere) simplification is also useful when dealing with objects such as [[Rotation matrix|rotational matrices]]. Spherical coordinates are also useful in analyzing systems that have some degree of symmetry about a point, including: [[multiple integral|volume integrals]] inside a sphere; the potential energy field surrounding a concentrated mass or charge; or global weather simulation in a planet's atmosphere.
[[Image:Bosch 36W column loudspeaker polar pattern.png|thumb|upright=1.2|The output pattern of the industrial [[loudspeaker]] shown here uses spherical polar plots taken at six frequencies]]

Three dimensional modeling of [[loudspeaker]] output patterns can be used to predict their performance. A number of polar plots are required, taken at a wide selection of frequencies, as the pattern changes greatly with frequency. Polar plots help to show that many loudspeakers tend toward omnidirectionality at lower frequencies.

An important application of spherical coordinates provides for the [[separation of variables]] in two [[partial differential equations]]{{mdash}}the [[Laplace equation|Laplace]] and the [[Helmholtz equation]]s{{mdash}}that arise in many physical problems. The angular portions of the solutions to such equations take the form of [[spherical harmonics]]. Another application is [[ergonomic design]], where {{mvar|r}} is the arm length of a stationary person and the angles describe the direction of the arm as it reaches out. The spherical coordinate system is also commonly used in 3D [[game development]] to rotate the camera around the player's position<ref>{{Cite web |title=Video Game Math: Polar and Spherical Notation |url=https://aie.edu/articles/video-game-math-polar-and-spherical-notation/ |access-date=2022-02-16 |website=Academy of Interactive Entertainment (AIE) |language=en-AU}}</ref>

=== In geography ===
{{main|Geographic coordinate system}}
{{see also|ECEF}}

Instead of inclination, the [[geographic coordinate system]] uses elevation angle (or ''[[latitude]]''), in the range (aka [[Interval (mathematics)|domain]]) {{math|−90° ≤ ''φ'' ≤ 90°}} and rotated north from the [[equator]] plane. Latitude (i.e., ''the angle'' of latitude) may be either ''[[geocentric latitude]]'', measured (rotated) from the Earth's center{{mdash}}and designated variously by {{math|''ψ'', ''q'', ''φ''′, ''φ''<sub>c</sub>, ''φ''<sub>g</sub>}}{{mdash}}or ''[[geodetic latitude]]'', measured (rotated) from the observer's [[local vertical]], and typically designated {{mvar|φ}}.
The polar angle (inclination), which is 90° minus the latitude and ranges from 0 to 180°, is called ''[[colatitude]]'' in geography.

The azimuth angle (or ''[[longitude]]'') of a given position on Earth, commonly denoted by {{mvar|λ}}, is measured in degrees east or west from some conventional reference [[meridian (geography)|meridian]] (most commonly the [[IERS Reference Meridian]]); thus its domain (or range) is {{math|−180° ≤ ''λ'' ≤ 180°}} and a given reading is typically designated "East" or "West". For positions on the [[Earth]] or other solid [[celestial body]], the reference plane is usually taken to be the plane perpendicular to the [[axis of rotation]]. <!--Must explain the conventions for positive latitude and longitude-->

Instead of the radial distance {{mvar|r}} geographers commonly use ''[[altitude]]'' above or below some local reference surface (''[[vertical datum]]''), which, for example, may be the [[mean sea level]]. When needed, the radial distance can be computed from the altitude by adding the [[radius of Earth]], which is approximately {{convert|6360|±|11|km|mi|0|abbr=in}}.

However, modern geographical coordinate systems are quite complex, and the positions implied by these simple formulae may be inaccurate by several kilometers. The precise standard meanings of ''latitude, longitude'' and ''altitude'' are currently defined by the [[World Geodetic System]] (WGS), and take into account the flattening of the Earth at the poles (about {{convert|21|km|mi|abbr=in|disp=or}}) and many other details.

[[Planetary coordinate system]]s use formulations analogous to the geographic coordinate system.

=== In astronomy ===
A series of [[astronomical coordinate systems]] are used to measure the elevation angle from several [[Fundamental plane (spherical coordinates)|fundamental planes]]. These reference planes include: 
the observer's [[Horizontal coordinate system|horizon]], the [[Galactic coordinate system|galactic equator]] (defined by the rotation of the [[Milky Way]]), the [[celestial equator]] (defined by Earth's rotation), the plane of the [[Ecliptic coordinate system|ecliptic]] (defined by Earth's orbit around the [[Sun]]), and the plane of the earth [[terminator (solar)|terminator]] (normal to the instantaneous direction to the [[Sun]]).