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Spherical coordinate system

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Coordinates comprising a distance and two angles

Thephysics convention. Spherical coordinates (r,θ,φ) as commonly used: (ISO 80000-2:2019): radial distancer (slant distance to origin), polar angleθ (theta) (angle with respect to positive polar axis), and azimuthal angleφ (phi) (angle of rotation from the initial meridian plane).This is the convention followed in this article.

Inmathematics, aspherical coordinate system specifies a given point inthree-dimensional space by using a distance and two angles as its threecoordinates. These are

theradial distancer along the line connecting the point to a fixed point called theorigin;
thepolar angleθ between this radial line and a givenpolar axis;^[a] and
theazimuthal angleφ, which is theangle of rotation of the radial line around the polar axis.^[b]

(See graphic regarding the "physics convention".)

Once the radius is fixed, the three coordinates (r,θ,φ), known as a 3-tuple, provide a coordinate system on asphere, typically called thespherical polar coordinates.Theplane passing through the origin andperpendicular to the polar axis (where the polar angle is aright angle) is called thereference plane (sometimesfundamental plane).

Terminology

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 theradius, orradial line, orradial coordinate. The polar angle may be calledinclination angle,zenith angle,normal angle, or thecolatitude. The user may choose to replace the inclination angle by itscomplement, theelevation angle (oraltitude angle), measured upward between the reference plane and the radial line—i.e., from the reference plane upward (towards to the positive z-axis) to the radial line. Thedepression angle is the negative of the elevation angle.(See graphic re the "physics convention"—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^[1] frequently encountered inphysics, where the naming tuple gives the order as: radial distance, polar angle, azimuthal angle, or $(r,\theta ,\varphi )$ . (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 $(\rho ,\theta ,\varphi )$ or $(r,\theta ,\varphi )$ —which switches the uses and meanings of symbols θ and φ. Other conventions may also be used, such asr for a radius from thez-axis that is not from the point of origin. Particular care must be taken to check the meaning of the symbols.

Themathematics convention. Spherical coordinates(r,θ,φ) as typically used: radial distancer, azimuthal angleθ, and polar angleφ. +The meanings ofθ andφ have been swapped—compared to thephysics convention. The 'south'-direction x-axis is depicted but the 'north'-direction x-axis is not. (As in physics,ρ (rho) is often used instead ofr to avoid confusion with the valuer in cylindrical and 2D polar coordinates.)

According to the conventions ofgeographical coordinate systems, positions are measured by latitude, longitude, and height (altitude). There are a number ofcelestial coordinate systems based on differentfundamental planes and with different terms for the various coordinates. The spherical coordinate systems used in mathematics normally useradians rather thandegrees; (note 90 degrees equalsπ⁄2 radians). And these systems of themathematics convention may measure the azimuthal anglecounterclockwise (i.e., from the south directionx-axis, or 180°, towards the east directiony-axis, or +90°)—rather than measureclockwise (i.e., from the north direction x-axis, or 0°, towards the east direction y-axis, or +90°), as done in thehorizontal coordinate system.^[2](See graphic re "mathematics convention".)

The spherical coordinate system of thephysics convention can be seen as a generalization of thepolar coordinate system inthree-dimensional space. It can be further extended to higher-dimensional spaces, and is then referred to as ahyperspherical coordinate system.

Definition

To define a spherical coordinate system, one must designate anorigin point in space,O, and two orthogonal directions: thezenith reference direction and theazimuth reference direction. These choices determine a reference plane that is typically defined as containing the point of origin and thex– and y–axes, either of which may be designated as theazimuth reference direction. The reference plane is perpendicular (orthogonal) to the zenith direction, and typically is designated "horizontal" to the zenith direction's "vertical". The spherical coordinates of a pointP then are defined as follows:

Theradius orradial distance is theEuclidean distance from the originO toP.
Theinclination (orpolar angle) is the signed angle from the zenith reference direction to the line segmentOP. (Elevation may be used as the polar angle instead ofinclination; see below.)
Theazimuth (orazimuthal angle) is the signed angle measured from theazimuth reference direction to the orthogonal projection of the radial line segmentOP on the reference plane.

The sign of the azimuth is determined by designating the rotation that is thepositive 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 (=π radians), the azimuth is arbitrary. If the radius is zero, both azimuth and inclination are arbitrary.)

Theelevation is the signed angle from the x-y reference plane to the radial line segmentOP, where positive angles are designated as upward, towards the zenith reference.Elevation is 90 degrees (=⁠π/2⁠ radians)minus inclination. Thus, if the inclination is 60 degrees (=⁠π/3⁠ radians), then the elevation is 30 degrees (=⁠π/6⁠ radians).

Inlinear algebra, thevector from the originO to the pointP is often called theposition vector ofP.

Conventions

Several different conventions exist for representing spherical coordinates and prescribing the naming order of their symbols. The 3-tuple number set $(r,\theta ,\varphi )$ denotes radial distance, the polar angle—"inclination", or as the alternative, "elevation"—and the azimuthal angle. It is the common practice within the physics convention, as specified byISO standard80000-2:2019, and earlier inISO 31-11 (1992).

As stated above, this article describes the ISO "physics convention"—unless otherwise noted.

However, some authors (including mathematicians) use the symbolρ (rho) for radius, or radial distance,φ for inclination (or elevation) andθ for azimuth—while others keep the use ofr for the radius; all which "provides a logical extension of the usual polar coordinates notation".^[3] As to order, some authors list the azimuthbefore the inclination (or the elevation) angle. Some combinations of these choices result in aleft-handed coordinate system. The standard "physics convention" 3-tuple set $(r,\theta ,\varphi )$ conflicts with the usual notation for two-dimensionalpolar coordinates and three-dimensionalcylindrical coordinates, whereθ is often used for the azimuth.^[3]

Angles are typically measured indegrees (°) or inradians (rad), where 360° = 2π rad. The use of degrees is most common in geography, astronomy, and engineering, where radians are commonly used in mathematics andtheoretical physics. The unit for radial distance is usually determined by the context, as occurs in applications of the 'unit sphere', seeapplications.

When the system is used to designate physical three-space, it is customary to assign positive to azimuth angles measured in thecounterclockwise sense from the reference direction on the reference plane—as seen from the "zenith" side of the plane. This convention is used in particular for geographical coordinates, where the "zenith" direction isnorth and the positive azimuth (longitude) angles are measured eastwards from someprime meridian.

Major conventions
coordinates set order	corresponding local geographical directions (Z,X,Y)	right/left-handed
(r,θ_inc,φ_az,right)	(U,S,E)	right
(r,φ_az,right,θ_el)	(U,E,N)	right
(r,θ_el,φ_az,right)	(U,N,E)	left

Note:Easting (E), Northing (N), Upwardness (U). In the case of(U,S,E) the localazimuth angle would be measuredcounterclockwise fromS toE.

Unique coordinates

Any spherical coordinate triplet (or tuple) $(r,\theta ,\varphi )$ 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 $(-r,\theta ,\varphi )$ , which is equivalent to $(r,\theta {+}180^{\circ },\varphi )$ or $(r,90^{\circ }{-}\theta ,\varphi {+}180^{\circ })$ for anyr,θ, andφ. Moreover, $(r,-\theta ,\varphi )$ is equivalent to $(r,\theta ,\varphi {+}180^{\circ })$ .

When necessary to define a unique set of spherical coordinates for each point, the user must restrict therange, aka interval, of each coordinate. A common choice is:

radial distance:r ≥ 0,
polar angle:0° ≤θ ≤ 180°, or0 rad ≤θ ≤π rad,
azimuth :0° ≤φ < 360°, or0 rad ≤φ < 2π rad.

But instead of the interval[0°, 360°), the azimuthφ is typically restricted to thehalf-open interval(−180°, +180°], or(−π, +π ] radians, which is the standard convention for geographic longitude.

For the polar angleθ, the range (interval) for inclination is[0°, 180°], which is equivalent to elevation range (interval)[−90°, +90°]. In geography, the latitude is the elevation.

Even with these restrictions, if the polar angle (inclination) is 0° or 180°—elevation is −90° or +90°—then the azimuth angle is arbitrary; and ifr 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(r,θ,φ), whereθ is inclination, the user would: mover units from the origin in the zenith reference direction (z-axis); then rotate by the amount of the azimuth angle (φ) about the originfrom the designatedazimuth reference direction, (i.e., either the x- or y-axis, seeDefinition, above); and then rotatefrom the z-axis by the amount of theθ angle.

Applications

In themathematics convention: A globe showing aunit sphere, withtuple coordinates of pointP (red): its radial distancer (red, not labeled); its azimuthal angleθ (not labeled); and its polar angle ofinclinationφ (not labeled). The radial distance upward along thezenith–axis from the point of origin to the surface of the sphere is assigned the value unity, or 1. + In this image,r appears to equal 4/6, or .67, (of unity); i.e., four of the six 'nested shells' to the surface. The azimuth angleθ appears to equal positive 90°, as rotatedcounterclockwise from the azimuth-reference x–axis; and the inclinationφ appears to equal 30°, as rotated from the zenith–axis. (Note the 'full' rotation, or inclination, from the zenith–axis to the y–axis is 90°).

Just as the two-dimensionalCartesian coordinate system is useful—has a wide set of applications—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 inCartesian coordinates with the equationx² +y² +z² =c² can be described inspherical coordinates by the simple equationr =c. (In this system—shown here in the mathematics convention—the sphere is adapted as aunit 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 asrotational matrices. Spherical coordinates are also useful in analyzing systems that have some degree of symmetry about a point, including:volume integrals inside a sphere; the potential energy field surrounding a concentrated mass or charge; or global weather simulation in a planet's atmosphere.

The output pattern of the industrialloudspeaker shown here uses spherical polar plots taken at six frequencies

Three dimensional modeling ofloudspeaker 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 theseparation of variables in twopartial differential equations—theLaplace and theHelmholtz equations—that arise in many physical problems. The angular portions of the solutions to such equations take the form ofspherical harmonics. Another application isergonomic design, wherer 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 3Dgame development to rotate the camera around the player's position^[4]

In geography

Main article:Geographic coordinate system

See also:ECEF

Instead of inclination, thegeographic coordinate system uses elevation angle (orlatitude), in the range (akadomain)−90° ≤φ ≤ 90° and rotated north from theequator plane. Latitude (i.e.,the angle of latitude) may be eithergeocentric latitude, measured (rotated) from the Earth's center—and designated variously byψ,q,φ′,φ_c,φ_g—orgeodetic latitude, measured (rotated) from the observer'slocal vertical, and typically designatedφ.The polar angle (inclination), which is 90° minus the latitude and ranges from 0 to 180°, is calledcolatitude in geography.

The azimuth angle (orlongitude) of a given position on Earth, commonly denoted byλ, is measured in degrees east or west from some conventional referencemeridian (most commonly theIERS Reference Meridian); thus its domain (or range) is−180° ≤λ ≤ 180° and a given reading is typically designated "East" or "West". For positions on theEarth or other solidcelestial body, the reference plane is usually taken to be the plane perpendicular to theaxis of rotation.

Instead of the radial distancer geographers commonly usealtitude above or below some local reference surface (vertical datum), which, for example, may be themean sea level. When needed, the radial distance can be computed from the altitude by adding theradius of Earth, which is approximately 6,360 ± 11 km (3,952 ± 7 miles).

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 oflatitude, longitude andaltitude are currently defined by theWorld Geodetic System (WGS), and take into account the flattening of the Earth at the poles (about 21 km or 13 miles) and many other details.

Planetary coordinate systems use formulations analogous to the geographic coordinate system.

In astronomy

A series ofastronomical coordinate systems are used to measure the elevation angle from severalfundamental planes. These reference planes include: the observer'shorizon, thegalactic equator (defined by the rotation of theMilky Way), thecelestial equator (defined byEarth's rotation), the plane of theecliptic (defined by Earth's orbit around theSun), and the plane of the earthterminator (normal to the instantaneous direction to theSun).

Coordinate system conversions

See 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: radiusr, inclinationθ, azimuthφ) can be obtained from itsCartesian coordinates(x,y,z) by the formulae

${\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arccos {\frac {z}{\sqrt {x^{2}+y^{2}+z^{2}}}}=\arccos {\frac {z}{r}}={\begin{cases}\arctan {\frac {\sqrt {x^{2}+y^{2}}}{z}}&{\text{if }}z>0\\\pi +\arctan {\frac {\sqrt {x^{2}+y^{2}}}{z}}&{\text{if }}z<0\\+{\frac {\pi }{2}}&{\text{if }}z=0{\text{ and }}{\sqrt {x^{2}+y^{2}}}\neq 0\\{\text{undefined}}&{\text{if }}x=y=z=0\\\end{cases}}\\\varphi &=\operatorname {sgn}(y)\arccos {\frac {x}{\sqrt {x^{2}+y^{2}}}}={\begin{cases}\arctan({\frac {y}{x}})&{\text{if }}x>0,\\\arctan({\frac {y}{x}})+\pi &{\text{if }}x<0{\text{ and }}y\geq 0,\\\arctan({\frac {y}{x}})-\pi &{\text{if }}x<0{\text{ and }}y<0,\\+{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y>0,\\-{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y<0,\\{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}\end{aligned}}$

Theinverse tangent denoted inφ = arctan⁠y/x⁠ must be suitably defined, taking into account the correct quadrant of(x,y), as done in the equations above. See the article onatan2.

Alternatively, the conversion can be considered as two sequentialrectangular to polar conversions: the first in the Cartesianxy plane from(x,y) to(R,φ), whereR is the projection ofr onto thexy-plane, and the second in the CartesianzR-plane from(z,R) to(r,θ). The correct quadrants forφ andθ 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 Cartesianxy plane, thatθ is inclination from thez direction, and that the azimuth angles are measured from the Cartesianx axis (so that they axis hasφ = +90°). Ifθ measures elevation from the reference plane instead of inclination from the zenith the arccos above becomes an arcsin, and thecosθ andsinθ below become switched.

Conversely, the Cartesian coordinates may be retrieved from the spherical coordinates (radiusr,inclinationθ,azimuthφ), wherer ∈[0, ∞),θ ∈[0,π],φ ∈[0, 2π), by ${\begin{aligned}x&=r\sin \theta \,\cos \varphi ,\\y&=r\sin \theta \,\sin \varphi ,\\z&=r\cos \theta .\end{aligned}}$

Cylindrical coordinates

Main article:Cylindrical coordinate system

Cylindrical coordinates (axialradiusρ,azimuthφ,elevationz) may be converted into spherical coordinates (central radiusr,inclinationθ,azimuthφ), by the formulas

${\begin{aligned}r&={\sqrt {\rho ^{2}+z^{2}}},\\\theta &=\arctan {\frac {\rho }{z}}=\arccos {\frac {z}{\sqrt {\rho ^{2}+z^{2}}}},\\\varphi &=\varphi .\end{aligned}}$

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

${\begin{aligned}\rho &=r\sin \theta ,\\\varphi &=\varphi ,\\z&=r\cos \theta .\end{aligned}}$

These formulae assume that the two systems have the same origin and same reference plane, measure the azimuth angleφ in the same senses from the same axis, and that the spherical angleθ is inclination from the cylindricalz axis.

Ellipsoidal coordinates

See also:Ellipsoidal coordinates

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

$ax^{2}+by^{2}+cz^{2}=d.$

The modified spherical coordinates of a point in P in the ISO convention (i.e. for physics:radiusr,inclinationθ,azimuthφ) can be obtained from itsCartesian coordinates(x,y,z) by the formulae

${\begin{aligned}x&={\frac {1}{\sqrt {a}}}r\sin \theta \,\cos \varphi ,\\y&={\frac {1}{\sqrt {b}}}r\sin \theta \,\sin \varphi ,\\z&={\frac {1}{\sqrt {c}}}r\cos \theta ,\\r^{2}&=ax^{2}+by^{2}+cz^{2}.\end{aligned}}$

An infinitesimal volume element is given by

$\mathrm {d} V=\left|{\frac {\partial (x,y,z)}{\partial (r,\theta ,\varphi )}}\right|\,dr\,d\theta \,d\varphi ={\frac {1}{\sqrt {abc}}}r^{2}\sin \theta \,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi ={\frac {1}{\sqrt {abc}}}r^{2}\,\mathrm {d} r\,\mathrm {d} \Omega .$

The square-root factor comes from the property of thedeterminant that allows a constant to be pulled out from a column:

${\begin{vmatrix}ka&b&c\\kd&e&f\\kg&h&i\end{vmatrix}}=k{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}.$

Integration and differentiation in spherical coordinates

Unit vectors in spherical coordinates

The following equations (Iyanaga 1977) assume that the colatitudeθ is the inclination from the positivez axis, as in thephysics convention discussed.

Theline element for an infinitesimal displacement from(r,θ,φ) to(r + dr,θ + dθ,φ + dφ) is $\mathrm {d} \mathbf {r} =\mathrm {d} r\,{\hat {\mathbf {r} }}+r\,\mathrm {d} \theta \,{\hat {\boldsymbol {\theta }}}+r\sin {\theta }\,\mathrm {d} \varphi \,\mathbf {\hat {\boldsymbol {\varphi }}} ,$ where ${\begin{aligned}{\hat {\mathbf {r} }}&=\sin \theta \cos \varphi \,{\hat {\mathbf {x} }}+\sin \theta \sin \varphi \,{\hat {\mathbf {y} }}+\cos \theta \,{\hat {\mathbf {z} }},\\{\hat {\boldsymbol {\theta }}}&=\cos \theta \cos \varphi \,{\hat {\mathbf {x} }}+\cos \theta \sin \varphi \,{\hat {\mathbf {y} }}-\sin \theta \,{\hat {\mathbf {z} }},\\{\hat {\boldsymbol {\varphi }}}&=-\sin \varphi \,{\hat {\mathbf {x} }}+\cos \varphi \,{\hat {\mathbf {y} }}\end{aligned}}$ are the local orthogonalunit vectors in the directions of increasingr,θ, andφ, respectively,andx̂,ŷ, andẑ are the unit vectors in Cartesian coordinates. The linear transformation to this right-handed coordinate triplet is arotation matrix, $R={\begin{pmatrix}\sin \theta \cos \varphi &\sin \theta \sin \varphi &{\hphantom {-}}\cos \theta \\\cos \theta \cos \varphi &\cos \theta \sin \varphi &-\sin \theta \\-\sin \varphi &\cos \varphi &{\hphantom {-}}0\end{pmatrix}}.$

This gives the transformation from the Cartesian to the spherical, the other way around is given by its inverse.Note: the matrix is anorthogonal matrix, that is, its inverse is simply itstranspose.

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

The general form of the formula to prove the differential line element, is^[5] $\mathrm {d} \mathbf {r} =\sum _{i}{\frac {\partial \mathbf {r} }{\partial x_{i}}}\,\mathrm {d} x_{i}=\sum _{i}\left|{\frac {\partial \mathbf {r} }{\partial x_{i}}}\right|{\frac {\frac {\partial \mathbf {r} }{\partial x_{i}}}{\left|{\frac {\partial \mathbf {r} }{\partial x_{i}}}\right|}}\,\mathrm {d} x_{i}=\sum _{i}\left|{\frac {\partial \mathbf {r} }{\partial x_{i}}}\right|\,\mathrm {d} x_{i}\,{\hat {\boldsymbol {x}}}_{i},$ that is, the change in $\mathbf {r}$ is decomposed into individual changes corresponding to changes in the individual coordinates.

To apply this to the present case, one needs to calculate how $\mathbf {r}$ changes with each of the coordinates. In the conventions used, $\mathbf {r} ={\begin{bmatrix}r\sin \theta \,\cos \varphi \\r\sin \theta \,\sin \varphi \\r\cos \theta \end{bmatrix}},x_{1}=r,x_{2}=\theta ,x_{3}=\varphi .$

Thus, ${\frac {\partial \mathbf {r} }{\partial r}}={\begin{bmatrix}\sin \theta \,\cos \varphi \\\sin \theta \,\sin \varphi \\\cos \theta \end{bmatrix}}=\mathbf {\hat {r}} ,\quad {\frac {\partial \mathbf {r} }{\partial \theta }}={\begin{bmatrix}r\cos \theta \,\cos \varphi \\r\cos \theta \,\sin \varphi \\-r\sin \theta \end{bmatrix}}=r\,{\hat {\boldsymbol {\theta }}},\quad {\frac {\partial \mathbf {r} }{\partial \varphi }}={\begin{bmatrix}-r\sin \theta \,\sin \varphi \\{\hphantom {-}}r\sin \theta \,\cos \varphi \\0\end{bmatrix}}=r\sin \theta \,\mathbf {\hat {\boldsymbol {\varphi }}} .$

The desired coefficients are the magnitudes of these vectors:^[5] $\left|{\frac {\partial \mathbf {r} }{\partial r}}\right|=1,\quad \left|{\frac {\partial \mathbf {r} }{\partial \theta }}\right|=r,\quad \left|{\frac {\partial \mathbf {r} }{\partial \varphi }}\right|=r\sin \theta .$

Thesurface element spanning fromθ toθ + dθ andφ toφ + dφ on a spherical surface at (constant) radiusr is then $\mathrm {d} S_{r}=\left\|{\frac {\partial {\mathbf {r} }}{\partial \theta }}\times {\frac {\partial {\mathbf {r} }}{\partial \varphi }}\right\|\mathrm {d} \theta \,\mathrm {d} \varphi =\left|r{\hat {\boldsymbol {\theta }}}\times r\sin \theta {\boldsymbol {\hat {\varphi }}}\right|\mathrm {d} \theta \,\mathrm {d} \varphi =r^{2}\sin \theta \,\mathrm {d} \theta \,\mathrm {d} \varphi ~.$

Thus the differentialsolid angle is $\mathrm {d} \Omega ={\frac {\mathrm {d} S_{r}}{r^{2}}}=\sin \theta \,\mathrm {d} \theta \,\mathrm {d} \varphi .$

The surface element in a surface of polar angleθ constant (a cone with vertex at the origin) is $\mathrm {d} S_{\theta }=r\sin \theta \,\mathrm {d} \varphi \,\mathrm {d} r.$

The surface element in a surface of azimuthφ constant (a vertical half-plane) is $\mathrm {d} S_{\varphi }=r\,\mathrm {d} r\,\mathrm {d} \theta .$

Thevolume element spanning fromr tor + dr,θ toθ + dθ, andφ toφ + dφ is specified by thedeterminant of theJacobian matrix ofpartial derivatives, $J={\frac {\partial (x,y,z)}{\partial (r,\theta ,\varphi )}}={\begin{pmatrix}\sin \theta \cos \varphi &r\cos \theta \cos \varphi &-r\sin \theta \sin \varphi \\\sin \theta \sin \varphi &r\cos \theta \sin \varphi &{\hphantom {-}}r\sin \theta \cos \varphi \\\cos \theta &-r\sin \theta &{\hphantom {-}}0\end{pmatrix}},$ namely $\mathrm {d} V=\left|{\frac {\partial (x,y,z)}{\partial (r,\theta ,\varphi )}}\right|\,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi =r^{2}\sin \theta \,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi =r^{2}\,\mathrm {d} r\,\mathrm {d} \Omega ~.$

Thus, for example, a functionf(r,θ,φ) can be integrated over every point inR³ by thetriple integral $\int \limits _{0}^{2\pi }\int \limits _{0}^{\pi }\int \limits _{0}^{\infty }f(r,\theta ,\varphi )r^{2}\sin \theta \,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi ~.$

Thedel operator in this system leads to the following expressions for thegradient andLaplacian for scalar fields, ${\begin{aligned}\nabla f&={\partial f \over \partial r}{\hat {\mathbf {r} }}+{1 \over r}{\partial f \over \partial \theta }{\hat {\boldsymbol {\theta }}}+{1 \over r\sin \theta }{\partial f \over \partial \varphi }{\hat {\boldsymbol {\varphi }}},\\[8pt]\nabla ^{2}f&={1 \over r^{2}}{\partial \over \partial r}\left(r^{2}{\partial f \over \partial r}\right)+{1 \over r^{2}\sin \theta }{\partial \over \partial \theta }\left(\sin \theta {\partial f \over \partial \theta }\right)+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2}f \over \partial \varphi ^{2}}\\[8pt]&=\left({\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial }{\partial r}}\right)f+{1 \over r^{2}\sin \theta }{\partial \over \partial \theta }\left(\sin \theta {\frac {\partial }{\partial \theta }}\right)f+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}f~,\\[8pt]\end{aligned}}$ And it leads to the following expressions for thedivergence andcurl ofvector fields,

$\nabla \cdot \mathbf {A} ={\frac {1}{r^{2}}}{\partial \over \partial r}\left(r^{2}A_{r}\right)+{\frac {1}{r\sin \theta }}{\partial \over \partial \theta }\left(\sin \theta A_{\theta }\right)+{\frac {1}{r\sin \theta }}{\partial A_{\varphi } \over \partial \varphi },$ ${\begin{aligned}\nabla \times \mathbf {A} ={}&{\frac {1}{r\sin \theta }}\left[{\partial \over \partial \theta }\left(A_{\varphi }\sin \theta \right)-{\partial A_{\theta } \over \partial \varphi }\right]{\hat {\mathbf {r} }}\\[4pt]&{}+{\frac {1}{r}}\left[{1 \over \sin \theta }{\partial A_{r} \over \partial \varphi }-{\partial \over \partial r}\left(rA_{\varphi }\right)\right]{\hat {\boldsymbol {\theta }}}\\[4pt]&{}+{\frac {1}{r}}\left[{\partial \over \partial r}\left(rA_{\theta }\right)-{\partial A_{r} \over \partial \theta }\right]{\hat {\boldsymbol {\varphi }}},\end{aligned}}$

Further, the inverse Jacobian in Cartesian coordinates is $J^{-1}={\begin{pmatrix}{\dfrac {x}{r}}&{\dfrac {y}{r}}&{\dfrac {z}{r}}\\\\{\dfrac {xz}{r^{2}{\sqrt {x^{2}+y^{2}}}}}&{\dfrac {yz}{r^{2}{\sqrt {x^{2}+y^{2}}}}}&{\dfrac {-\left(x^{2}+y^{2}\right)}{r^{2}{\sqrt {x^{2}+y^{2}}}}}\\\\{\dfrac {-y}{x^{2}+y^{2}}}&{\dfrac {x}{x^{2}+y^{2}}}&0\end{pmatrix}}.$ Themetric tensor in the spherical coordinate system is $g=J^{T}J$ .

Distance in spherical coordinates

In spherical coordinates, given two points withφ being the azimuthal coordinate ${\begin{aligned}{\mathbf {r} }&=(r,\theta ,\varphi ),\\{\mathbf {r} '}&=(r',\theta ',\varphi ')\end{aligned}}$ The distance between the two points can be expressed as^[6] ${\begin{aligned}{\mathbf {D} }&={\sqrt {r^{2}+r'^{2}-2rr'(\sin {\theta }\sin {\theta '}\cos {(\varphi -\varphi ')}+\cos {\theta }\cos {\theta '})}}\end{aligned}}$

Kinematics

In spherical coordinates, the position of a point or particle (although better written as atriple $(r,\theta ,\varphi )$ ) can be written as^[7] $\mathbf {r} =r\mathbf {\hat {r}} .$ Its velocity is then^[7] $\mathbf {v} ={\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}={\dot {r}}\mathbf {\hat {r}} +r\,{\dot {\theta }}\,{\hat {\boldsymbol {\theta }}}+r\,{\dot {\varphi }}\sin \theta \,\mathbf {\hat {\boldsymbol {\varphi }}}$ and its acceleration is^[7] ${\begin{aligned}\mathbf {a} ={}&{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}\\[1ex]={}&{\hphantom {+}}\;\left({\ddot {r}}-r\,{\dot {\theta }}^{2}-r\,{\dot {\varphi }}^{2}\sin ^{2}\theta \right)\mathbf {\hat {r}} \\&{}+\left(r\,{\ddot {\theta }}+2{\dot {r}}\,{\dot {\theta }}-r\,{\dot {\varphi }}^{2}\sin \theta \cos \theta \right){\hat {\boldsymbol {\theta }}}\\&{}+\left(r{\ddot {\varphi }}\,\sin \theta +2{\dot {r}}\,{\dot {\varphi }}\,\sin \theta +2r\,{\dot {\theta }}\,{\dot {\varphi }}\,\cos \theta \right){\hat {\boldsymbol {\varphi }}}\end{aligned}}$

The angular momentum is $\mathbf {L} =\mathbf {r} \times \mathbf {p} =\mathbf {r} \times m\mathbf {v} =mr^{2}\left(-{\dot {\varphi }}\sin \theta \,\mathbf {\hat {\boldsymbol {\theta }}} +{\dot {\theta }}\,{\hat {\boldsymbol {\varphi }}}\right)$ Where $m {\displaystyle m}$ is mass. In the case of a constantφ or elseθ =⁠π/2⁠, this reduces tovector calculus in polar coordinates.

The corresponding angular momentum operator then follows from the phase-space reformulation of the above, $\mathbf {L} =-i\hbar ~\mathbf {r} \times \nabla =i\hbar \left({\frac {\hat {\boldsymbol {\theta }}}{\sin(\theta )}}{\frac {\partial }{\partial \phi }}-{\hat {\boldsymbol {\phi }}}{\frac {\partial }{\partial \theta }}\right).$

The torque is given as^[7] $\mathbf {\tau } ={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}=\mathbf {r} \times \mathbf {F} =-m\left(2r{\dot {r}}{\dot {\varphi }}\sin \theta +r^{2}{\ddot {\varphi }}\sin {\theta }+2r^{2}{\dot {\theta }}{\dot {\varphi }}\cos {\theta }\right){\hat {\boldsymbol {\theta }}}+m\left(r^{2}{\ddot {\theta }}+2r{\dot {r}}{\dot {\theta }}-r^{2}{\dot {\varphi }}^{2}\sin \theta \cos \theta \right){\hat {\boldsymbol {\varphi }}}$

Thekinetic energy is given as^[7] $E_{k}={\frac {1}{2}}m\left[\left({\dot {r}}\right)^{2}+\left(r{\dot {\theta }}\right)^{2}+\left(r{\dot {\varphi }}\sin \theta \right)^{2}\right]$

See also

Celestial coordinate system – System for specifying positions of celestial objectsPages displaying short descriptions of redirect targets
Coordinate system – Method for specifying point positions
Del in cylindrical and spherical coordinates – Mathematical gradient operator in certain coordinate systems
Double Fourier sphere method – Mathematical technique
Elevation (ballistics) – Angle in ballistics
Euler angles – Description of the orientation of a rigid body
Gimbal lock – Loss of one degree of freedom in a three-dimensional, three-gimbal mechanism
Hypersphere – Mathematical objectPages displaying short descriptions of redirect targets
Jacobian matrix and determinant – Matrix of partial derivatives of a vector-valued function
List of canonical coordinate transformations
Sphere – Set of points equidistant from a center
Spherical harmonic – Special mathematical functions defined on the surface of a spherePages displaying short descriptions of redirect targets
Theodolite – Optical surveying instrument
Vector fields in cylindrical and spherical coordinates – Vector field representation in 3D curvilinear coordinate systems
Yaw, pitch, and roll – Principal directions in aviationPages displaying short descriptions of redirect targets

Notes

^Anoriented line, so the polar angle is anoriented angle reckoned from the polar axis maindirection, not its opposite direction.
^If the polar axis is made to coincide with positivez-axis, the azimuthal angleφ may be calculated as the angle between either of thex-axis ory-axis and theorthogonal projection of the radial line onto the referencex-y-plane — which isorthogonal to thez-axis and passes through the fixed point of origin, completing a three-dimensionalCartesian coordinate system.

References

^"ISO 80000-2:2019 Quantities and units – Part 2: Mathematics".ISO. 19 May 2020. pp. 20–21. Item no. 2-17.3. Retrieved2020-08-12.
^Duffett-Smith, P and Zwart, J, p. 34.
^^a ^bEric W. Weisstein (2005-10-26)."Spherical Coordinates".MathWorld. Retrieved2010-01-15.
^"Video Game Math: Polar and Spherical Notation".Academy of Interactive Entertainment (AIE). Retrieved2022-02-16.
^^a ^b"Line element (dl) in spherical coordinates derivation/diagram".Stack Exchange. October 21, 2011.
^"Distance between two points in spherical coordinates".
^^a ^b ^c ^d ^eReed, Bruce Cameron (2019).Keplerian ellipses : the physics of the gravitational two-body problem. Morgan & Claypool Publishers, Institute of Physics. San Rafael [California] (40 Oak Drive, San Rafael, CA, 94903, US).ISBN 978-1-64327-470-6.OCLC 1104053368.{{cite book}}: CS1 maint: location (link) CS1 maint: location missing publisher (link)

Bibliography

Iyanaga, Shōkichi; Kawada, Yukiyosi (1977).Encyclopedic Dictionary of Mathematics. MIT Press.ISBN 978-0262090162.
Morse PM,Feshbach H (1953).Methods of Theoretical Physics, Part I. New York: McGraw-Hill. p. 658.ISBN 0-07-043316-X.LCCN 52011515.{{cite book}}:ISBN / Date incompatibility (help)
Margenau H, Murphy GM (1956).The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 177–178.LCCN 55010911.
Korn GA,Korn TM (1961).Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. pp. 174–175.LCCN 59014456. ASIN B0000CKZX7.
Sauer R, Szabó I (1967).Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. pp. 95–96.LCCN 67025285.
Moon P, Spencer DE (1988). "Spherical Coordinates (r, θ, ψ)".Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer-Verlag. pp. 24–27 (Table 1.05).ISBN 978-0-387-18430-2.
Duffett-Smith P, Zwart J (2011).Practical Astronomy with your Calculator or Spreadsheet, 4th Edition. New York: Cambridge University Press. p. 34.ISBN 978-0521146548.

External links

v t e Orthogonal coordinate systems
Two dimensional	Cartesian Polar (Log-polar) Parabolic Bipolar Elliptic
Three dimensional	Cartesian Cylindrical Spherical Parabolic Paraboloidal Oblate spheroidal Prolate spheroidal Ellipsoidal Elliptic cylindrical Toroidal Bispherical Bipolar cylindrical Conical 6-sphere

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