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{{Short description|3-dimensional coordinate system}}

[[File:3D Spherical.svg|thumb|240x240px|The '''''physics convention'''''. Spherical coordinates {{math|(''r'', ''θ'', ''φ'')}} as commonly used: (ISO 80000-2:2019): radial distance {{mvar|r}} (slant distance to origin), polar angle {{mvar|θ}} (theta) (angle with respect to positive polar axis), and azimuthal angle {{mvar|φ}} (phi) (angle of rotation from the initial meridian plane). '''''This is the convention followed in this article.''''']]

In [[Mathematics|mathematics]], a '''spherical coordinate system''' is a [[Coordinate system|coordinate system]] for [[Three-dimensional space|three-dimensional space]] where the position of a given point in space is specified by ''three'' numbers, ('''r''',  '''θ''',  '''φ'''): the ''radial distance'' of the ''radial line'' '''r''' connecting the point to the fixed point of origin (which is located on a fixed polar axis, or [[Zenith|zenith]] direction axis, or '''z'''-axis); the ''polar angle  '''θ''''' of the ''radial line '''r'''; and the ''azimuthal angle '''φ''''' of the ''radial line '''r'''''.

The ''polar angle '''θ''''' is measured ''between'' the '''z'''-axis ''and'' the radial line '''r'''.    The ''azimuthal angle '''φ''''' is measured ''between'' the   orthogonal projection   of the radial line '''r''' onto  the reference '''x-y'''-plane{{mdash}}which is [[Orthogonality|orthogonal]] to the '''z'''-axis and passes through the fixed point of origin{{mdash}}''and'' either of the fixed '''x'''-axis or '''y'''-axis, both of which are orthogonal to the '''z'''-axis and to each other. (See graphic re the "physics convention".) <!-- Please maintain the bolding of symbols (in this lead paragraph only) for ease of distinguishing same while introducing them to reader. --> 

Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-[[Tuple|tuple]], provide a coordinate system on a [[Sphere|sphere]], typically called the '''spherical polar coordinates'''. ''Nota bene: the physics convention is followed in this article; (See both graphics re "physics convention" and re "mathematics convention")''.

The radial distance from the fixed point of origin is also called the ''radius'', or ''radial line'', or ''radial coordinate''. The polar angle may be called ''inclination angle'', ''zenith angle'', ''normal angle'', or the ''[[Colatitude|colatitude]]''. The user may choose to ignore the ''inclination angle'' and use the ''elevation angle'' instead, which is measured upward between the reference plane and the radial line{{mdash}}i.e., from the reference plane upward (towards to the positive z-axis) to the radial line. The ''depression angle'' is the negative of the elevation angle. ''(See graphic re the "physics convention"{{mdash}}not "mathematics convention".)''

Both the use of symbols and the naming order of tuple coordinates differ among the several sources and disciplines. This article will use the ISO convention<ref>{{Cite web |title=ISO 80000-2:2019 Quantities and units – Part 2: Mathematics |url=https://www.iso.org/standard/64973.html |access-date=2020-08-12 |website=ISO |date=19 May 2020 |pages=20–21 |language=en |id=Item no. 2-17.3}}</ref> frequently encountered in ''physics'', where the naming tuple gives the order as: radial distance, polar angle, azimuthal angle, or ''<math>(r,\theta,\varphi)</math>''. (See graphic re the "physics convention".) ''In contrast, the conventions in many mathematics books and texts give the naming order differently as: radial  distance, "azimuthal angle", "polar angle", and <math>(\rho,\theta,\varphi)</math> or <math>(r,\theta,\varphi)</math>{{mdash}}which switches the uses and meanings of symbols θ and φ''. Other conventions may also be used, such as ''r'' for a radius from the ''z-''axis that is not from the point of origin. ''Particular care must be taken to check the meaning of the symbols''. <!-- Please maintain a consistent convention in this article. -->

[[File:3D Spherical 2.svg|thumb|240px|right|The '''''mathematics convention'''''. Spherical coordinates {{math|(''r'', ''θ'', ''φ'')}} as typically used: radial distance {{mvar|r}}, azimuthal angle {{mvar|θ}}, and polar angle {{mvar|φ}}. + ''The meanings of {{mvar|θ}} and {{mvar|φ}} have been swapped''{{mdash}}compared to the '''physics convention'''. The 'south'-direction x-axis is depicted but the 'north'-direction x-axis is not. (As in physics, {{mvar|ρ}} (rho) is often used instead of {{mvar|r}} to avoid confusion with the value {{mvar|r}} in cylindrical and 2D polar coordinates.)]]

According to the conventions of geographical coordinate systems, positions are measured by latitude, longitude, and height (altitude). There are a number of [[Astronomy:Celestial coordinate system|celestial coordinate system]]s based on different [[Fundamental plane (spherical coordinates)|fundamental planes]] and with different terms for the various coordinates. The spherical coordinate systems used in mathematics normally use [[Radian|radian]]s rather than [[Degree (angle)|degrees]]; (note 90 degrees equals π<sub>/2</sub> radians). And these systems of the  ''mathematics convention'' may measure the azimuthal angle ''counterclockwise'' (i.e., from the south direction {{mvar|x}}-axis, or 180°, towards the east direction {{mvar|y}}-axis, or +90°){{mdash}}rather than measure ''clockwise'' (i.e., from the north direction x-axis, or 0°, towards the east direction y-axis, or +90°), as done in the [[Horizontal coordinate system|horizontal coordinate system]].<ref>Duffett-Smith, P and Zwart, J, p. 34.</ref> ''(See graphic re "mathematics convention".)''

The spherical coordinate system of the ''physics convention'' can be seen as a generalization of the [[Polar coordinate system|polar coordinate system]] in [[Three-dimensional space|three-dimensional space]]. 
It can be further extended to higher-dimensional spaces, and is then referred to as a [[N-sphere#Spherical coordinates|''hyperspherical coordinate system'']].

== Definition ==
To define a spherical coordinate system, one must designate an ''origin'' point in space, ''{{mvar|O}}'', and two orthogonal directions: the ''zenith reference'' direction and the ''azimuth reference'' direction. These choices determine a reference plane that is typically defined as containing the point of origin and the ''x{{ndash}} and y{{ndash}}axes'', either of which may be designated as the ''azimuth reference'' direction. The reference plane is perpendicular (orthogonal) to the zenith direction, and typically is desiginated "horizontal" to the zenith direction's "vertical". The spherical coordinates of a point {{mvar|P}} then are defined as follows:

* The ''radius'' or ''radial distance'' is the [[Euclidean distance]] from the origin ''{{mvar|O}}'' to ''{{mvar|P}}''.
* The ''inclination'' (or ''polar angle'') is the signed angle from the zenith reference direction to the line segment {{mvar|OP}}. (''Elevation'' may be  used as the polar angle instead of ''inclination''; see below.)
* The ''azimuth'' (or ''azimuthal angle'') is the signed angle measured from the ''azimuth reference'' direction to the orthogonal projection of the radial line segment {{mvar|OP}} on the reference plane.
The sign of the azimuth is determined by designating the rotation that is the ''positive'' sense of turning about the zenith. This choice is arbitrary, and is part of the coordinate system definition. 
(If the inclination is either zero or 180 degrees (= {{pi}} radians), the azimuth is arbitrary. If the radius is zero, both azimuth and inclination are arbitrary.)

The ''elevation'' is the signed angle from the x-y reference plane to the radial line segment {{mvar|OP}}, where positive angles are designated as upward, towards the zenith reference. ''Elevation'' is 90 degrees (= {{sfrac|{{pi}}|2}} radians) ''minus inclination''. Thus, if the inclination is 60 degrees (= {{sfrac|{{pi}}|3}} radians), then the elevation is 30 degrees (= {{sfrac|{{pi}}|6}} radians).

In [[Linear algebra|linear algebra]], the [[Euclidean vector|vector]] from the origin {{mvar|O}} to the point {{mvar|P}} is often called the ''position vector'' of ''P''.

=== Conventions ===
Several different conventions exist for representing spherical coordinates and prescribing the naming order of their symbols. The 3-tuple number set <math>(r,\theta,\varphi)</math> denotes radial distance, the polar angle{{mdash}}"inclination", or as the alternative, "elevation"{{mdash}}and the azimuthal angle. It is the common practice within the physics convention, as specified by [[Organization:International Organization for Standardization|ISO]] standard 80000-2:2019, and earlier in [[ISO 31-11]] (1992). 

''As stated above, this article describes the ISO "physics convention"{{mdash}}unless otherwise noted.''

However, some authors (including mathematicians) use the symbol ''ρ'' (rho) for radius, or radial distance, ''φ'' for inclination (or elevation) and ''θ'' for azimuth{{mdash}}while others keep the use of ''r'' for the radius; all which "provides a logical extension of the usual polar coordinates notation".<ref name="http://mathworld.wolfram.com/SphericalCoordinates.html">{{cite web |url=http://mathworld.wolfram.com/SphericalCoordinates.html |title=Spherical Coordinates |author=[[Biography:Eric W. Weisstein|Eric W. Weisstein]] |publisher=[[MathWorld]] |date=2005-10-26 |access-date=2010-01-15}}</ref> As to order, some authors list the azimuth ''before'' the inclination (or the elevation) angle. Some combinations of these choices result in a [[Physics:Right-hand rule|left-handed]] coordinate system. The standard "physics convention" 3-tuple set <math>(r,\theta,\varphi)</math> conflicts with the usual notation for two-dimensional [[Polar coordinate system|polar coordinates]] and three-dimensional [[Cylindrical coordinate system|cylindrical coordinates]], where {{mvar|θ}} is often used for the azimuth.<ref name="http://mathworld.wolfram.com/SphericalCoordinates.html" />

Angles are typically measured in [[Degree (angle)|degrees]] (°) or in [[Radian|radian]]s (rad), where 360°&nbsp;=&nbsp;2{{pi}} rad. The use of degrees is most common in geography, astronomy, and engineering, where radians are commonly used in mathematics and theoretical physics. The unit for radial distance is usually determined by the context, as occurs in applications of the 'unit sphere', see #Applications.

When the system is used to designate physical three-space, it is customary to assign positive to azimuth angles measured in the ''counterclockwise'' sense from the reference direction on the reference plane{{mdash}}as seen from the "zenith" side of the plane. This convention is used in particular for geographical coordinates, where the "zenith" direction is [[North|north]] and the positive azimuth (longitude) angles are measured eastwards from some [[Earth:Prime meridian|prime meridian]].

{| class="wikitable" style="text-align:center"
|+ Major conventions
|-
! coordinates set order !! corresponding local geographical directions <br/> {{math|(''Z'', ''X'', ''Y'')}} !! right/left-handed
|-
| {{math|(''r'', ''θ''<sub>inc</sub>, ''φ''<sub>az,right</sub>)}} || {{math|(''U'', ''S'', ''E'')}} || right
|-
| {{math|(''r'', ''φ''<sub>az,right</sub>, ''θ''<sub>el</sub>)}}|| {{math|(''U'', ''E'', ''N'')}} || right
|-
| {{math|(''r'', ''θ''<sub>el</sub>, ''φ''<sub>az,right</sub>)}}|| {{math|(''U'', ''N'', ''E'')}} || left
|}
'''Note:''' [[Easting and northing|Easting ({{mvar|E}}), Northing ({{mvar|N}})]], Upwardness ({{mvar|U}}). In the case of {{math|(''U'', ''S'', ''E'')}} the local [[Azimuth|azimuth]] angle would be measured counterclockwise from {{mvar|S}} to {{mvar|E}}.

=== Unique coordinates ===
Any spherical coordinate triplet (or tuple) <math>(r,\theta,\varphi)</math> specifies a single point of three-dimensional space. On the reverse view, any single point has infinitely many equivalent spherical coordinates. That is, the user can add or subtract any number of full turns to the angular measures without changing the angles themselves, and therefore without changing the point. It is convenient in many contexts to use negative radial distances, the convention being <math>(-r,-\theta,\varphi)</math>, which is equivalent to <math>(r,\theta,\varphi)</math> for any {{mvar|r}}, {{mvar|θ}}, and {{mvar|φ}}. Moreover, <math>(r,-\theta,\varphi)</math> is equivalent to <math>(r,\theta,\varphi{+}180^\circ)</math>.

When necessary to define a unique set of spherical coordinates for each point, the user must restrict the [[Interval (mathematics)|range, aka interval]], of each coordinate. A common choice is:

{{startplainlist|indent=1}}
* radial distance: {{math|''r'' ≥ 0,}}
* polar angle: {{math|0° ≤ ''θ'' ≤ 180°}}, or {{math|0 rad ≤ ''θ'' ≤ {{pi}} rad}},
* azimuth : {{math|0° ≤  ''φ'' < 360°}}, or {{math|0 rad ≤  ''φ'' < 2{{pi}} rad}}. 

But instead of the interval {{closed-open|0°, 360°}}, the azimuth {{mvar|φ}} is typically restricted to the [[Interval (mathematics)#Definitions|half-open interval]] {{open-closed|−180°, +180°}}, or {{open-closed|−{{pi}}, +{{pi}} }} radians, which is the standard convention for geographic longitude.

For the polar angle {{mvar|θ}}, the range (interval) for inclination is {{closed-closed|0°, 180°}}, which is equivalent to elevation range (interval) {{closed-closed|−90°, +90°}}. In geography, the latitude is the elevation.

Even with these restrictions, if the polar angle (inclination) is 0° or 180°{{mdash}}elevation is  −90° or +90°{{mdash}}then the azimuth angle is arbitrary; and if {{mvar|r}} is zero, both azimuth and polar angles are arbitrary. To define the coordinates as unique, the user can assert the convention that (in these cases) the arbitrary coordinates are set to zero.

=== Plotting ===
To plot any dot from its spherical coordinates {{math|(''r'', ''θ'', ''φ'')}}, where {{mvar|θ}} is inclination, the user would: move {{mvar|r}} units from the origin in the zenith reference direction (z-axis); then rotate by the amount of the azimuth angle ({{mvar|φ}}) about the origin  ''from'' the  designated ''azimuth reference'' direction, (i.e., either the x{{ndash}} or y{{ndash}}axis, see Definition, above); and then rotate ''from'' the z-axis by the amount of the {{mvar|θ}} angle.

== Applications ==
[[File:Spherical coordinate system.svg|thumb|upright=1.2|right|In the '''''mathematics convention''''': A globe showing a [[Unit sphere|unit sphere]], with [[Tuple|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|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 [[Engineering:Loudspeaker|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|separation of variables]] in two partial differential equations{{mdash}}the 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 [[Physics:Spherical harmonics|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|Earth:Geographic coordinate system}}

Instead of inclination, the [[Earth:Geographic coordinate system|geographic coordinate system]] uses elevation angle (or ''[[Earth:Latitude|latitude]]''), in the range (aka [[Interval (mathematics)|domain]]) {{math|−90° ≤ ''φ'' ≤ 90°}} and rotated north from the [[Astronomy:Equator|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|colatitude]]'' in geography.

The azimuth angle (or ''[[Earth:Longitude|longitude]]'') of a given position on Earth, commonly denoted by {{mvar|λ}}, is measured in degrees east or west from some conventional reference 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 [[Astronomy:Earth|Earth]] or other solid celestial body, the reference plane is usually taken to be the plane perpendicular to the [[Physics:Axis of rotation|axis of rotation]]. <!--Must explain the conventions for positive latitude and longitude-->

Instead of the radial distance {{mvar|r}} geographers commonly use ''[[Altitude|altitude]]'' above or below some local reference surface (''[[Vertical datum|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.

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

=== In astronomy ===
A series of [[Astronomy:Astronomical coordinate systems|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 [[Astronomy:Milky Way|Milky Way]]), the [[Celestial equator|celestial equator]] (defined by Earth's rotation), the plane of the [[Astronomy:Ecliptic coordinate system|ecliptic]] (defined by Earth's orbit around the [[Astronomy:Sun|Sun]]), and the plane of the earth [[Astronomy:Terminator (solar)|terminator]] (normal to the instantaneous direction to the [[Astronomy:Sun|Sun]]).

== Coordinate system conversions ==
{{also|List of common coordinate transformations#To spherical coordinates}}
As the spherical coordinate system is only one of many three-dimensional coordinate systems, there exist equations for converting coordinates between the spherical coordinate system and others.

=== Cartesian coordinates ===
The spherical coordinates of a point in the ISO convention (i.e. for physics:  radius  {{mvar|r}}, inclination  {{mvar|θ}}, azimuth  {{mvar|φ}}) can be obtained from its [[Cartesian coordinate system|Cartesian coordinates]] {{math|(''x'', ''y'', ''z'')}} by the formulae

<math display="block">\begin{align}
r &= \sqrt{x^2 + y^2 + z^2} \\
\theta &= \arccos\frac{z}{\sqrt{x^2 + y^2 + z^2}} = \arccos\frac{z}{r}=
\begin{cases}
 \arctan\frac{\sqrt{x^2+y^2}}{z} &\text{if } z > 0 \\
 \pi +\arctan\frac{\sqrt{x^2+y^2}}{z} &\text{if } z < 0 \\
 +\frac{\pi}{2} &\text{if } z = 0 \text{ and } xy \neq 0 \\
 \text{undefined} &\text{if } x=y=z = 0 \\
\end{cases} \\
\varphi &= \sgn(y)\arccos\frac{x}{\sqrt{x^2+y^2}} =
\begin{cases}
 \arctan(\frac{y}{x}) &\text{if } x > 0, \\
 \arctan(\frac{y}{x}) + \pi &\text{if } x < 0 \text{ and } y \geq 0, \\
 \arctan(\frac{y}{x}) - \pi &\text{if } x < 0 \text{ and } y < 0, \\
 +\frac{\pi}{2} &\text{if } x = 0 \text{ and } y > 0, \\
 -\frac{\pi}{2} &\text{if } x = 0 \text{ and } y < 0, \\
 \text{undefined} &\text{if } x = 0 \text{ and } y = 0.
\end{cases}
\end{align}</math>

The inverse tangent denoted in {{math|''φ'' {{=}} arctan {{sfrac|''y''|''x''}}}} must be suitably defined, taking into account the correct quadrant of {{math|(''x'', ''y'')}}. See the article on atan2.

Alternatively, the conversion can be considered as two sequential [[Polar coordinate system#Converting between polar and Cartesian coordinates|rectangular to polar conversions]]: the first in the Cartesian {{mvar|xy}} plane from {{math|(''x'', ''y'')}} to {{math|(''R'', ''φ'')}}, where {{mvar|R}} is the projection of {{mvar|r}} onto the {{mvar|xy}}-plane, and the second in the Cartesian {{mvar|zR}}-plane from {{math|(''z'', ''R'')}} to {{math|(''r'', ''θ'')}}. The correct quadrants for {{mvar|φ}} and {{mvar|θ}} are implied by the correctness of the planar rectangular to polar conversions.

These formulae assume that the two systems have the same origin, that the spherical reference plane is the Cartesian {{mvar|xy}} plane, that {{mvar|θ}} is inclination from the {{mvar|z}} direction, and that the azimuth angles are measured from the Cartesian {{mvar|x}} axis (so that the {{mvar|y}} axis has {{math|''φ'' {{=}} +90°}}). If ''θ'' measures elevation from the reference plane instead of inclination from the zenith the arccos above becomes an arcsin, and the {{math|cos ''θ''}} and {{math|sin ''θ''}} below become switched.

Conversely, the Cartesian coordinates may be retrieved from the spherical coordinates (''radius'' {{mvar|r}}, ''inclination'' {{mvar|θ}}, ''azimuth'' {{mvar|φ}}), where {{math|''r'' ∈ {{closed-open|0, ∞}}}}, {{math|''θ'' ∈ {{closed-closed|0, {{pi}}}}}}, {{math|''φ'' ∈ {{closed-open|0, 2{{pi}}}}}}, by
<math display="block">\begin{align}
 x &= r \sin\theta \, \cos\varphi, \\
 y &= r \sin\theta \, \sin\varphi, \\
 z &= r \cos\theta.
\end{align}</math>

=== Cylindrical coordinates ===
{{main|Cylindrical coordinate system}}
<!-- if you notice--rho and r are described opposite of convention here--someone with extra time could remedy this -->

[[Cylindrical coordinate system|Cylindrical coordinates]] (''axial'' ''radius'' ''ρ'', ''azimuth'' <!--radians--> ''φ'', ''elevation'' ''z'') may be converted into spherical coordinates (''central radius'' ''r'', ''inclination'' ''θ'', ''azimuth'' ''φ''), by the formulas

<math display="block">\begin{align}
 r &= \sqrt{\rho^2 + z^2}, \\
 \theta &= \arctan\frac{\rho}{z} = \arccos\frac{z}{\sqrt{\rho^2 + z^2}}, \\
 \varphi &= \varphi.
\end{align}</math>

Conversely, the spherical coordinates may be converted into cylindrical coordinates by the formulae

<math display="block">\begin{align}
 \rho &= r \sin \theta, \\
 \varphi &= \varphi, \\
 z &= r \cos \theta.
\end{align}</math>

These formulae assume that the two systems have the same origin and same reference plane, measure the azimuth angle {{mvar|φ}} in the same senses from the same axis, and that the spherical angle {{mvar|θ}} is inclination from the cylindrical {{mvar|z}} axis.

== Generalization ==
It is also possible to deal with ellipsoids in Cartesian coordinates by using a modified version of the spherical coordinates. 

Let P be an ellipsoid specified by the level set

<math display="block">ax^2 + by^2 + cz^2 = d.</math>

The modified spherical coordinates of a point in P in the ISO convention (i.e. for physics: ''radius'' {{mvar|r}}, ''inclination'' {{mvar|θ}}, ''azimuth'' {{mvar|φ}}) can be obtained from its [[Cartesian coordinate system|Cartesian coordinates]] {{math|(''x'', ''y'', ''z'')}} by the formulae

<math display="block">\begin{align}
 x &= \frac{1}{\sqrt{a}} r \sin\theta \, \cos\varphi, \\
 y &= \frac{1}{\sqrt{b}} r \sin\theta \, \sin\varphi, \\
 z &= \frac{1}{\sqrt{c}} r \cos\theta, \\
 r^{2} &= ax^2 + by^2 + cz^2.
\end{align}</math>

An infinitesimal volume element is given by

<math display="block">
\mathrm{d}V = \left|\frac{\partial(x, y, z)}{\partial(r, \theta, \varphi)}\right| \, dr\,d\theta\,d\varphi =
 \frac{1}{\sqrt{abc}} r^2 \sin \theta \,\mathrm{d}r \,\mathrm{d}\theta \,\mathrm{d}\varphi =
 \frac{1}{\sqrt{abc}} r^2 \,\mathrm{d}r \,\mathrm{d}\Omega.
</math>

The square-root factor comes from the property of the [[Determinant|determinant]] that allows a constant to be pulled out from a column:

<math display="block">
\begin{vmatrix}
 ka & b & c \\
 kd & e & f \\
 kg & h & i
\end{vmatrix} =
k \begin{vmatrix}
 a & b & c \\
 d & e & f \\
 g & h & i
\end{vmatrix}.
</math>

== Integration and differentiation in spherical coordinates ==
[[File:Kugelkoord-lokb-e.svg|thumb|240px|right|Unit vectors in spherical coordinates]]

The following equations (Iyanaga 1977) assume that the colatitude {{mvar|θ}} is the inclination from the positive {{mvar|z}} axis, as in the ''physics convention'' discussed.

The [[Line element|line element]] for an infinitesimal displacement from {{math|(''r'', ''θ'', ''φ'')}} to {{math|(''r'' + d''r'', ''θ'' + d''θ'', ''φ'' + d''φ'')}} is
<math display="block">
\mathrm{d}\mathbf{r} = \mathrm{d}r\,\hat{\mathbf r} + r\,\mathrm{d}\theta \,\hat{\boldsymbol\theta } + r \sin{\theta} \, \mathrm{d}\varphi\,\mathbf{\hat{\boldsymbol\varphi}},</math>
where
<math display="block">\begin{align}
 \hat{\mathbf r} &= \sin \theta \cos \varphi \,\hat{\mathbf x} +
  \sin \theta \sin \varphi \,\hat{\mathbf y} + \cos \theta \,\hat{\mathbf z}, \\
 \hat{\boldsymbol\theta} &= \cos \theta \cos \varphi \,\hat{\mathbf x} +
  \cos \theta \sin \varphi \,\hat{\mathbf y} - \sin \theta \,\hat{\mathbf z}, \\
 \hat{\boldsymbol\varphi} &= - \sin \varphi \,\hat{\mathbf x} +
  \cos \varphi \,\hat{\mathbf y}
\end{align}</math>
are the local orthogonal unit vectors in the directions of increasing {{mvar|r}}, {{mvar|θ}}, and {{mvar|φ}}, respectively,
and {{math|'''x̂'''}}, {{math|'''ŷ'''}}, and {{math|'''ẑ'''}} are the unit vectors in Cartesian coordinates. The linear transformation to this right-handed coordinate triplet is a [[Rotation matrix|rotation matrix]],
<math display="block">R = \begin{pmatrix}
     \sin\theta\cos\varphi&\sin\theta\sin\varphi&\hphantom{-}\cos\theta\\
     \cos\theta\cos\varphi&\cos\theta\sin\varphi&-\sin\theta\\
    -\sin\varphi&\cos\varphi   &\hphantom{-}0
   \end{pmatrix}.
</math>

This gives the transformation from the spherical to the cartesian, the other way around is given by its inverse.
Note: the matrix is an [[Orthogonal matrix|orthogonal matrix]], that is, its inverse is simply its [[Transpose|transpose]].

The Cartesian unit vectors are thus related to the spherical unit vectors by:
<math display="block">\begin{bmatrix}\mathbf{\hat x} \\ \mathbf{\hat y}  \\ \mathbf{\hat z} \end{bmatrix}
  = \begin{bmatrix} \sin\theta\cos\varphi & \cos\theta\cos\varphi & -\sin\varphi \\
                    \sin\theta\sin\varphi & \cos\theta\sin\varphi & \hphantom{-}\cos\varphi \\
                    \cos\theta         & -\sin\theta        & \hphantom{-}0 \end{bmatrix}
    \begin{bmatrix} \boldsymbol{\hat{r}} \\ \boldsymbol{\hat\theta} \\ \boldsymbol{\hat\varphi} \end{bmatrix}</math>

The general form of the formula to prove the differential line element, is<ref name="q74503">{{cite web |title=Line element (dl) in spherical coordinates derivation/diagram |date=October 21, 2011 |work=[[Company:Stack Exchange|Stack Exchange]] |url=https://math.stackexchange.com/q/74503}}</ref>
<math display="block">\mathrm{d}\mathbf{r} =
 \sum_i \frac{\partial \mathbf{r}}{\partial x_i} \,\mathrm{d}x_i =
 \sum_i \left|\frac{\partial \mathbf{r}}{\partial x_i}\right|
  \frac{\frac{\partial \mathbf{r}}{\partial x_i}}{\left|\frac{\partial \mathbf{r}}{\partial x_i}\right|} \, \mathrm{d}x_i =
 \sum_i \left|\frac{\partial \mathbf{r}}{\partial x_i}\right| \,\mathrm{d}x_i \, \hat{\boldsymbol{x}}_i,
</math>
that is, the change in <math>\mathbf r</math> is decomposed into individual changes corresponding to changes in the individual coordinates. 

To apply this to the present case, one needs to calculate how <math>\mathbf r</math> changes with each of the coordinates. In the conventions used, 
<math display="block">\mathbf{r} = \begin{bmatrix}
 r \sin\theta \, \cos\varphi \\
 r \sin\theta \, \sin\varphi \\
 r \cos\theta
\end{bmatrix}.</math>

Thus,
<math display="block">
\frac{\partial\mathbf r}{\partial r} = \begin{bmatrix}
 \sin\theta \, \cos\varphi \\
 \sin\theta \, \sin\varphi \\
 \cos\theta
\end{bmatrix}=\mathbf{\hat r}, \quad
\frac{\partial\mathbf r}{\partial \theta} = \begin{bmatrix}
 r \cos\theta \, \cos\varphi \\
 r \cos\theta \, \sin\varphi \\
 -r \sin\theta
\end{bmatrix}=r\,\hat{\boldsymbol\theta }, \quad
\frac{\partial\mathbf r}{\partial \varphi} = \begin{bmatrix}
 -r \sin\theta \, \sin\varphi \\
 \hphantom{-}r \sin\theta \, \cos\varphi \\
 0
\end{bmatrix}
=
r \sin\theta\,\mathbf{\hat{\boldsymbol\varphi}}  .
</math>

The desired coefficients are the magnitudes of these vectors:<ref name="q74503" />
<math display="block">
\left|\frac{\partial\mathbf r}{\partial r}\right| = 1, \quad
\left|\frac{\partial\mathbf r}{\partial \theta}\right| = r, \quad
\left|\frac{\partial\mathbf r}{\partial \varphi}\right| = r \sin\theta.
</math>

The [[Surface integral|surface element]] spanning from {{mvar|θ}} to {{math|''θ'' + d''θ''}} and {{mvar|φ}} to {{math|''φ'' + d''φ''}} on a spherical surface at (constant) radius {{mvar|r}} is then
<math display="block">
\mathrm{d}S_r =
 \left\|\frac{\partial {\mathbf r}}{\partial \theta} \times \frac{\partial {\mathbf r}}{\partial \varphi}\right\| \mathrm{d}\theta \,\mathrm{d}\varphi =
\left|r {\hat \boldsymbol\theta} \times r \sin \theta {\boldsymbol\hat \varphi} \right|= 
 r^2 \sin\theta \,\mathrm{d}\theta \,\mathrm{d}\varphi  ~.
</math>

Thus the differential [[Solid angle|solid angle]] is
<math display="block">\mathrm{d}\Omega = \frac{\mathrm{d}S_r}{r^2} = \sin\theta \,\mathrm{d}\theta \,\mathrm{d}\varphi.</math>

The surface element in a surface of polar angle {{mvar|θ}} constant (a cone with vertex the origin) is
<math display="block">\mathrm{d}S_\theta = r \sin\theta \,\mathrm{d}\varphi \,\mathrm{d}r.</math>

The surface element in a surface of azimuth {{mvar|φ}} constant (a vertical half-plane) is
<math display="block">\mathrm{d}S_\varphi = r \,\mathrm{d}r \,\mathrm{d}\theta.</math>

The [[Volume element|volume element]] spanning from {{mvar|r}} to {{math|''r'' + d''r''}}, {{mvar|θ}} to {{math|''θ'' + d''θ''}}, and {{mvar|φ}} to {{math|''φ'' + d''φ''}} is specified by the [[Determinant|determinant]] of the Jacobian matrix of [[Partial derivative|partial derivative]]s, 
<math display="block">
J =\frac{\partial(x,y,z)}{\partial(r,\theta,\varphi)}
  =\begin{pmatrix}
     \sin\theta\cos\varphi & r\cos\theta\cos\varphi & -r\sin\theta\sin\varphi\\
     \sin\theta\sin\varphi & r\cos\theta\sin\varphi & \hphantom{-}r\sin\theta\cos\varphi\\
     \cos\theta & -r\sin\theta & \hphantom{-}0
   \end{pmatrix},
</math>
namely 
<math display="block">
\mathrm{d}V = \left|\frac{\partial(x, y, z)}{\partial(r, \theta, \varphi)}\right| \,\mathrm{d}r \,\mathrm{d}\theta \,\mathrm{d}\varphi=
 r^2 \sin\theta \,\mathrm{d}r \,\mathrm{d}\theta \,\mathrm{d}\varphi =
 r^2 \,\mathrm{d}r \,\mathrm{d}\Omega ~.
</math>

Thus, for example, a function {{math|''f''(''r'', ''θ'', ''φ'')}} can be integrated over every point in {{math|'''R'''<sup>3</sup>}} by the [[Multiple integral#Spherical coordinates|triple integral]]
<math display="block">\int\limits_0^{2\pi} \int\limits_0^\pi \int\limits_0^\infty f(r, \theta, \varphi) r^2 \sin\theta \,\mathrm{d}r \,\mathrm{d}\theta \,\mathrm{d}\varphi ~.</math>

The [[Del|del]] operator in this system leads to the following expressions for the [[Gradient|gradient]] and Laplacian for scalar fields,
<math display="block">\begin{align}
\nabla f &= {\partial f \over \partial r}\hat{\mathbf r}
  + {1 \over r}{\partial f \over \partial \theta}\hat{\boldsymbol\theta}
  + {1 \over r\sin\theta}{\partial f \over \partial \varphi}\hat{\boldsymbol\varphi},  \\[8pt]
\nabla^2 f &= {1 \over r^2}{\partial \over \partial r} \left(r^2 {\partial f \over \partial r}\right) + {1 \over r^2 \sin\theta}{\partial \over \partial \theta} \left(\sin\theta {\partial f \over \partial \theta}\right)
+ {1 \over r^2 \sin^2\theta}{\partial^2 f \over \partial \varphi^2} \\[8pt]
& = \left(\frac{\partial^2}{\partial r^2} + \frac{2}{r} \frac{\partial}{\partial r}\right) f + {1 \over r^2 \sin\theta}{\partial \over \partial \theta} \left(\sin\theta \frac{\partial}{\partial \theta}\right) f + \frac{1}{r^2 \sin^2\theta}\frac{\partial^2}{\partial \varphi^2}f ~, \\[8pt]
\end{align}</math>And it leads to the following expressions for the [[Divergence|divergence]] and [[Curl (mathematics)|curl]] of [[Vector field|vector fields]],

<math display="block">\nabla \cdot \mathbf{A}
  = \frac{1}{r^2}{\partial \over \partial r}\left( r^2 A_r \right)
  + \frac{1}{r \sin\theta}{\partial \over \partial\theta} \left( \sin\theta A_\theta \right)
  + \frac{1}{r \sin \theta} {\partial A_\varphi \over \partial \varphi},</math><math display="block">\begin{align}
\nabla \times \mathbf{A} = {}
&      \frac{1}{r\sin\theta} \left[{\partial \over \partial \theta} \left( A_\varphi\sin\theta \right)
    - {\partial A_\theta \over \partial \varphi}\right] \hat{\mathbf r} \\[4pt]
& {} + \frac 1 r \left[{1 \over \sin\theta}{\partial A_r \over \partial \varphi}
    - {\partial \over \partial r} \left( r A_\varphi \right) \right] \hat{\boldsymbol\theta} \\[4pt]
& {} + \frac 1 r \left[{\partial \over \partial r} \left( r A_\theta \right)
    - {\partial A_r \over \partial \theta}\right] \hat{\boldsymbol\varphi},
\end{align}</math>

Further, the inverse Jacobian in Cartesian coordinates is
<math display="block">J^{-1} = \begin{pmatrix}
  \dfrac{x}{r}&\dfrac{y}{r}&\dfrac{z}{r}\\\\
  \dfrac{xz}{r^2\sqrt{x^2+y^2}}&\dfrac{yz}{r^2\sqrt{x^2+y^2}}&\dfrac{-\left(x^2 + y^2\right)}{r^2\sqrt{x^2+y^2}}\\\\
  \dfrac{-y}{x^2+y^2}&\dfrac{x}{x^2+y^2}&0
\end{pmatrix}.</math>
The [[Physics:Metric tensor|metric tensor]] in the spherical coordinate system is <math>g = J^T J </math>.

== Distance in spherical coordinates ==
In spherical coordinates, given two points with {{mvar|φ}} being the azimuthal coordinate
<math display="block">\begin{align}
 {\mathbf r} &= (r,\theta,\varphi), \\
 {\mathbf r'} &= (r',\theta',\varphi')
\end{align}</math>
The distance between the two points can be expressed as
<math display="block">\begin{align}
 {\mathbf D} &= \sqrt{r^2+r'^2-2rr'(\sin{\theta}\sin{\theta'}\cos{(\varphi-\varphi')} + \cos{\theta}\cos{\theta'})}
\end{align}</math>

== Kinematics ==
In spherical coordinates, the position of a point or particle (although better written as a [[Tuple|triple]]<math>(r,\theta, \varphi)</math>) can be written as<ref name="Cameron2019">{{Cite book |last=Reed |first=Bruce Cameron |url=https://www.worldcat.org/oclc/1104053368 |title=Keplerian ellipses : the physics of the gravitational two-body problem |date=2019 |others=Morgan & Claypool Publishers, Institute of Physics |isbn=978-1-64327-470-6 |location=San Rafael [California] (40 Oak Drive, San Rafael, CA, 94903, US) |oclc=1104053368}}</ref>
<math display="block">\mathbf{r} = r \mathbf{\hat r} .</math>
Its velocity is then<ref name="Cameron2019" />
<math display="block">\mathbf{v} = \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} = \dot{r} \mathbf{\hat r} + r\,\dot\theta\,\hat{\boldsymbol\theta } + r\,\dot\varphi \sin\theta\,\mathbf{\hat{\boldsymbol\varphi}}</math>
and its acceleration is<ref name="Cameron2019" />
<math display="block">
\begin{align}
\mathbf{a} = {} & \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} \\[1ex]
= {} & \hphantom{+}\; \left( \ddot{r} - r\,\dot\theta^2 - r\,\dot\varphi^2\sin^2\theta \right)\mathbf{\hat r} \\
& {} + \left( r\,\ddot\theta + 2\dot{r}\,\dot\theta - r\,\dot\varphi^2\sin\theta\cos\theta \right) \hat{\boldsymbol\theta } \\
& {} + \left( r\ddot\varphi\,\sin\theta + 2\dot{r}\,\dot\varphi\,\sin\theta + 2 r\,\dot\theta\,\dot\varphi\,\cos\theta \right) \hat{\boldsymbol\varphi}
\end{align}
</math>

The [[Physics:Angular momentum#Orbital angular momentum in three dimensions|angular momentum]] is 
<math display="block">  \mathbf{L} =
\mathbf{r} \times \mathbf{p} = \mathbf{r} \times m\mathbf{v} = m r^2 \left(- \dot\varphi \sin\theta\,\mathbf{\hat{\boldsymbol\theta}} + \dot\theta\,\hat{\boldsymbol\varphi }\right) 
</math>
Where <math>m</math> is mass. In the case of a constant {{mvar|φ}} or else {{math|''θ'' {{=}} {{sfrac|{{pi}}|2}}}}, this reduces to [[Polar coordinate system#Vector calculus|vector calculus in polar coordinates]].

The corresponding [[Physics:Angular momentum operator#Orbital angular momentum in spherical coordinates|angular momentum operator]] then follows from the phase-space reformulation of the above,  
<math display="block"> \mathbf{L}= -i\hbar ~\mathbf{r} \times \nabla =i \hbar \left(\frac{\hat{\boldsymbol{\theta}}}{\sin(\theta)} \frac{\partial}{\partial\phi} - \hat{\boldsymbol{\phi}} \frac{\partial}{\partial\theta}\right). </math>

The torque is given as<ref name="Cameron2019" />
<math display="block">  \mathbf{\tau} =
\frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t} = \mathbf{r} \times \mathbf{F} = -m \left(2r\dot{r}\dot{\varphi}\sin\theta + r^2\ddot{\varphi}\sin{\theta} + 2r^2\dot{\theta}\dot{\varphi}\cos{\theta} \right)\hat{\boldsymbol\theta} + m \left(r^2\ddot{\theta} + 2r\dot{r}\dot{\theta} - r^2\dot{\varphi}^2\sin\theta\cos\theta \right) \hat{\boldsymbol\varphi}
</math>

The kinetic energy is given as<ref name="Cameron2019" />
<math display="block"> E_k =
\frac{1}{2}m \left[ \left(\dot{r}^2\right) + \left(r\dot{\theta}\right)^2 + \left(r\dot{\varphi}\sin\theta\right)^2 \right]
</math>

== See also ==

* {{Annotated link |Celestial coordinate system}}
* {{Annotated link |Coordinate system}}
* {{Annotated link |Del in cylindrical and spherical coordinates}}
* {{Annotated link |Double Fourier sphere method}}
* {{Annotated link |Elevation (ballistics)}}
* {{Annotated link |Euler angles}}
* {{Annotated link |Gimbal lock}}
* {{Annotated link |Hypersphere}}
* {{Annotated link |Jacobian matrix and determinant}}
* {{Annotated link |List of canonical coordinate transformations}}
* {{Annotated link |Sphere}}
* {{Annotated link |Spherical harmonic}}
* {{Annotated link |Theodolite}}
* {{Annotated link |Vector fields in cylindrical and spherical coordinates}}
* {{Annotated link |Yaw, pitch, and roll}}

== Notes ==
{{reflist}}

== Bibliography ==
* {{cite book |last1=Iyanaga |first1=Shōkichi |last2=Kawada |first2=Yukiyosi |title=[[Encyclopedic Dictionary of Mathematics]] |date=1977 |publisher=MIT Press |isbn=978-0262090162}}
* {{cite book |author=[[Biography:Philip M. Morse|Morse PM]], [[Biography:Herman Feshbach|Feshbach H]] |year=1953 |title=Methods of Theoretical Physics, Part I |publisher=McGraw-Hill |location=New York |isbn=0-07-043316-X |pages=658 |lccn=52011515}}
* {{cite book |author=[[Biography:Henry Margenau|Margenau H]], Murphy GM |year=1956 |title=The Mathematics of Physics and Chemistry |url=https://archive.org/details/mathematicsofphy0002marg |url-access=registration |publisher=D. van Nostrand |location=New York |pages=[https://archive.org/details/mathematicsofphy0002marg/page/177 177–178] |lccn=55010911}}
* {{cite book |author=Korn GA, [[Biography:Theresa M. Korn|Korn TM]] |year=1961 |title=Mathematical Handbook for Scientists and Engineers |publisher=McGraw-Hill |location=New York |id=ASIN B0000CKZX7 |pages=174–175 |lccn=59014456}}
* {{cite book |author=Sauer R, Szabó I |year=1967 |title=Mathematische Hilfsmittel des Ingenieurs |publisher=Springer Verlag |location=New York |pages=95–96 |lccn=67025285}}
* {{cite book |author=Moon P, Spencer DE |year=1988 |chapter=Spherical Coordinates (r, θ, ψ) |title=Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions |edition=corrected 2nd ed., 3rd print |publisher=Springer-Verlag |location=New York |pages=24–27 (Table 1.05) |isbn=978-0-387-18430-2}}
* {{cite book |author=Duffett-Smith P, Zwart J |year=2011 |title=Practical Astronomy with your Calculator or Spreadsheet, 4th Edition |publisher=Cambridge University Press |location=New York |pages=34 |isbn=978-0521146548}}

== External links ==
* {{springer|title=Spherical coordinates|id=p/s086660}}
* [http://mathworld.wolfram.com/SphericalCoordinates.html MathWorld description of spherical coordinates]
* [http://www.random-science-tools.com/maths/coordinate-converter.htm Coordinate Converter – converts between polar, Cartesian and spherical coordinates]
 
{{Orthogonal coordinate systems}}


[[Category:Orthogonal coordinate systems]]
[[Category:Three-dimensional coordinate systems]]
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{{Sourceattribution|Spherical coordinate system}}