Spherical Polar Coordinates
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a spherical coordinate system is a
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
for
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
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 The zenith (, ) is an imaginary point directly "above" a particular location, on the celestial sphere. "Above" means in the vertical direction (plumb line) opposite to the gravity direction at that location (nadir). The zenith is the "highest" ...
direction, and the ''
azimuthal angle An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematicall ...
'' of its
orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
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 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 (analogous to the or ...
. The radial distance is also called the ''radius'' or ''radial coordinate''. The polar angle may be called '' colatitude'', ''
zenith angle The zenith (, ) is an imaginary point directly "above" a particular location, on the celestial sphere. "Above" means in the vertical direction ( plumb line) opposite to the gravity direction at that location ( nadir). The zenith is the "highe ...
'', '' normal angle'', or ''
inclination angle Orbital inclination measures the tilt of an object's orbit around a celestial body. It is expressed as the angle between a reference plane and the orbital plane or axis of direction of the orbiting object. For a satellite orbiting the Earth ...
''. When radius is fixed, the two angular coordinates make a coordinate system on the
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
sometimes called spherical polar coordinates. The use of symbols and the order of the coordinates differs among sources and disciplines. This article will use the ISO convention frequently encountered in physics: ''(r,\theta,\varphi)'' gives the radial distance, polar angle, and azimuthal angle. By contrast, in many mathematics books, (\rho,\theta,\varphi) or (r,\theta,\varphi) gives the radial distance, azimuthal angle, and polar angle, switching the meanings of ''θ'' and ''φ''. Other conventions are also used, such as ''r'' for radius from the ''z-''axis, so great care needs to be taken to check the meaning of the symbols. According to the conventions of geographical coordinate systems, positions are measured by latitude, longitude, and height (altitude). There are a number of
celestial coordinate system 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 (e.g. the true hor ...
s based on different fundamental planes and with different terms for the various coordinates. The spherical coordinate systems used in mathematics normally use
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s rather than degrees and measure the azimuthal angle counterclockwise from the -axis to the -axis rather than clockwise from north (0°) to east (+90°) like the horizontal coordinate system. The polar angle is often replaced by the ''elevation angle'' measured from the reference plane towards the positive Z axis, so that the elevation angle of zero is at the horizon; the ''depression angle'' is the negative of the elevation angle. The spherical coordinate system generalizes the two-dimensional polar coordinate system. It can also be extended to higher-dimensional spaces and is then referred to as a hyperspherical coordinate system.


Definition

To define a spherical coordinate system, one must choose two orthogonal directions, the ''zenith'' and the ''azimuth reference'', and an ''origin'' point in space. These choices determine a reference plane that contains the origin and is perpendicular to the zenith. The spherical coordinates of a point are then defined as follows: * The ''radius'' or ''radial distance'' is the
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
from the origin to . * The ''azimuth'' (or ''azimuthal angle'') is the signed angle measured from the azimuth reference direction to the orthogonal projection of the line segment on the reference plane. * The ''inclination'' (or ''polar angle'') is the angle between the zenith direction and the line segment . The sign of the azimuth is determined by choosing what is a ''positive'' sense of turning about the zenith. This choice is arbitrary, and is part of the coordinate system's definition. The ''elevation'' angle is the signed angle between the reference plane and the line segment , where positive angles are oriented towards the zenith. Equivalently, it is 90 degrees ( radians) minus the inclination angle. If the inclination is zero or 180 degrees ( radians), the azimuth is arbitrary. If the radius is zero, both azimuth and inclination are arbitrary. In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
, the
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
from the origin to the point is often called the '' position vector'' of ''P''.


Conventions

Several different conventions exist for representing the three coordinates, and for the order in which they should be written. The use of (r,\theta,\varphi) to denote radial distance, inclination (or elevation), and azimuth, respectively, is common practice in physics, and is specified by
ISO ISO is the most common abbreviation for the International Organization for Standardization. ISO or Iso may also refer to: Business and finance * Iso (supermarket), a chain of Danish supermarkets incorporated into the SuperBest chain in 2007 * Iso ...
standard 80000-2:2019, and earlier in
ISO 31-11 ISO 31-11:1992 was the part of international standard ISO 31 that defines ''mathematical signs and symbols for use in physical sciences and technology''. It was superseded in 2009 by ISO 80000-2:2009 and subsequently revised in 2019 as ISO-800 ...
(1992). However, some authors (including mathematicians) use ''ρ'' for radial distance, ''φ'' for inclination (or elevation) and ''θ'' for azimuth, and ''r'' for radius from the ''z-''axis, which "provides a logical extension of the usual polar coordinates notation". Some authors may also list the azimuth before the inclination (or elevation). Some combinations of these choices result in a
left-handed In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subjecti ...
coordinate system. The standard convention (r,\theta,\varphi) conflicts with the usual notation for two-dimensional
polar coordinates 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 (analogous to the or ...
and three-dimensional
cylindrical coordinates 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 d ...
, where is often used for the azimuth. The angles are typically measured in degrees (°) or
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s (rad), where 360° = 2 rad. Degrees are most common in geography, astronomy, and engineering, whereas radians are commonly used in mathematics and theoretical physics. The unit for radial distance is usually determined by the context. When the system is used for physical three-space, it is customary to use positive sign for azimuth angles that are measured in the counter-clockwise 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 is
north North is one of the four compass points or cardinal directions. It is the opposite of south and is perpendicular to east and west. ''North'' is a noun, adjective, or adverb indicating Direction (geometry), direction or geography. Etymology T ...
and positive azimuth (longitude) angles are measured eastwards from some
prime meridian A prime meridian is an arbitrary meridian (a line of longitude) in a geographic coordinate system at which longitude is defined to be 0°. Together, a prime meridian and its anti-meridian (the 180th meridian in a 360°-system) form a great c ...
. : ::Note: easting (), northing (), upwardness (). Local
azimuth An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematicall ...
angle would be measured, e.g.,
counterclockwise Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...
from to in the case of .


Unique coordinates

Any spherical coordinate triplet (r,\theta,\varphi) specifies a single point of three-dimensional space. On the other hand, every point has infinitely many equivalent spherical coordinates. One can add or subtract any number of full turns to either angular measure without changing the angles themselves, and therefore without changing the point. It is also convenient, in many contexts, to allow negative radial distances, with the convention that (-r,-\theta,\varphi180^\circ) is equivalent to (r,\theta,\varphi) for any , , and . Moreover, (r,-\theta,\varphi) is equivalent to (r,\theta,\varphi180^\circ). If it is necessary to define a unique set of spherical coordinates for each point, one must restrict their ranges. A common choice is : : : However, the azimuth is often restricted to the interval , or in radians, instead of . This is the standard convention for geographic longitude. For , the range for inclination is equivalent to for elevation. In geography, the latitude is the elevation. Even with these restrictions, if is 0° or 180° (elevation is 90° or −90°) then the azimuth angle is arbitrary; and if is zero, both azimuth and inclination/elevation are arbitrary. To make the coordinates unique, one can use the convention that in these cases the arbitrary coordinates are zero.


Plotting

To plot a dot from its spherical coordinates , where is inclination, move units from the origin in the zenith direction, rotate by about the origin towards the azimuth reference direction, and rotate by about the zenith in the proper direction.


Applications

Just as the two-dimensional
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
is useful on the plane, a two-dimensional spherical coordinate system is useful on the surface of a sphere. In this system, the sphere is taken as a
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit b ...
, so the radius is unity and can generally be ignored. This simplification can also be very useful when dealing with objects such as rotational matrices. Spherical coordinates are useful in analyzing systems that have some degree of symmetry about a point, such as volume integrals inside a sphere, the potential energy field surrounding a concentrated mass or charge, or global weather simulation in a planet's atmosphere. A sphere that has the Cartesian equation has the simple equation in spherical coordinates. Two important
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
that arise in many physical problems, Laplace's equation and the Helmholtz equation, allow a
separation of variables In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
in spherical coordinates. The angular portions of the solutions to such equations take the form of
spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a ...
. Another application is ergonomic design, where is the arm length of a stationary person and the angles describe the direction of the arm as it reaches out. Three dimensional modeling of
loudspeaker A loudspeaker (commonly referred to as a speaker or speaker driver) is an electroacoustic transducer that converts an electrical audio signal into a corresponding sound. A ''speaker system'', also often simply referred to as a "speaker" or " ...
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. The spherical coordinate system is also commonly used in 3D
game development Video game development (or gamedev) is the software development, process of developing a video game. The effort is undertaken by a video game developer, developer, ranging from a single person to an international team dispersed across the globe. ...
to rotate the camera around the player's position


In geography

To a first approximation, the
geographic coordinate system The geographic coordinate system (GCS) is a spherical or ellipsoidal coordinate system for measuring and communicating positions directly on the Earth as latitude and longitude. It is the simplest, oldest and most widely used of the various ...
uses elevation angle (''
latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
'') in degrees north of the
equator The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can als ...
plane, in the range , instead of inclination. Latitude is either ''
geocentric latitude In geography, latitude is a coordinate that specifies the north–south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pole ...
'', measured at the Earth's center and designated variously by or ''
geodetic latitude Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a '' reference ellipsoid''. They include geodetic latitude (north/south) , ''longitude'' (east/west) , and ellipsoidal height (also known as ...
'', measured by the observer's
local vertical In astronomy, geography, and related sciences and contexts, a '' direction'' or '' plane'' passing by a given point is said to be vertical if it contains the local gravity direction at that point. Conversely, a direction or plane is said to be ho ...
, and commonly designated . The polar angle, which is 90° minus the latitude and ranges from 0 to 180°, is called '' colatitude'' in geography. The azimuth angle (''
longitude Longitude (, ) is a geographic coordinate that specifies the east–west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter l ...
''), commonly denoted by , is measured in degrees east or west from some conventional reference
meridian Meridian or a meridian line (from Latin ''meridies'' via Old French ''meridiane'', meaning “midday”) may refer to Science * Meridian (astronomy), imaginary circle in a plane perpendicular to the planes of the celestial equator and horizon * ...
(most commonly the
IERS Reference Meridian The IERS Reference Meridian (IRM), also called the International Reference Meridian, is the prime meridian (0° longitude) maintained by the International Earth Rotation and Reference Systems Service (IERS). It passes about 5.3 arcseconds east ...
), so its domain is . For positions on the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
or other solid celestial body, the reference plane is usually taken to be the plane perpendicular to the axis of rotation. Instead of the radial distance, geographers commonly use ''
altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
'' above or below some reference surface (''
vertical datum In geodesy, surveying, hydrography and navigation, vertical datum or altimetric datum, is a reference coordinate surface used for vertical positions, such as the elevations of Earth-bound features (terrain, bathymetry, water level, and built stru ...
''), which may be the
mean sea level There are several kinds of mean in mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. ...
. The radial distance can be computed from the altitude by adding the
radius of Earth Earth radius (denoted as ''R''🜨 or R_E) is the distance from the center of Earth to a point on or near its surface. Approximating the figure of Earth by an Earth spheroid, the radius ranges from a maximum of nearly (equatorial radius, deno ...
, which is approximately . However, modern geographical coordinate systems are quite complex, and the positions implied by these simple formulae may be wrong by several kilometers. The precise standard meanings of latitude, longitude and altitude are currently defined by the
World Geodetic System The World Geodetic System (WGS) is a standard used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also descri ...
(WGS), and take into account the flattening of the Earth at the poles (about ) and many other details.
Planetary coordinate system A planetary coordinate system is a generalization of the geographic coordinate system and the geocentric coordinate system for planets other than Earth. Similar coordinate systems are defined for other solid celestial bodies, such as in the ''sele ...
s use formulations analogous to the geographic coordinate system.


In astronomy

A series of
astronomical coordinate systems 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 (e.g. the true horizo ...
are used to measure the elevation angle from different fundamental planes. These reference planes are the observer's
horizon The horizon is the apparent line that separates the surface of a celestial body from its sky when viewed from the perspective of an observer on or near the surface of the relevant body. This line divides all viewing directions based on whether i ...
, the
celestial equator The celestial equator is the great circle of the imaginary celestial sphere on the same plane as the equator of Earth. This plane of reference bases the equatorial coordinate system. In other words, the celestial equator is an abstract proj ...
(defined by Earth's rotation), the plane of the
ecliptic The ecliptic or ecliptic plane is the orbital plane of the Earth around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic again ...
(defined by Earth's orbit around the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
), the plane of the earth terminator (normal to the instantaneous direction to the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
), and the
galactic equator The galactic coordinate system is a celestial coordinate system in spherical coordinates, with the Sun as its center, the primary direction aligned with the approximate center of the Milky Way Galaxy, and the fundamental plane parallel to an ap ...
(defined by the rotation of the
Milky Way The Milky Way is the galaxy that includes our Solar System, with the name describing the galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars that cannot be individually distinguished by the naked eye ...
).


Coordinate system conversions

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 , inclination , azimuth ) can be obtained from its Cartesian coordinates by the formulae : \begin r &= \sqrt \\ \theta &= \arccos\frac = \arccos\frac= \begin \arctan\frac &\text z > 0 \\ \pi +\arctan\frac &\text z < 0 \\ +\frac &\text z = 0 \text xy \neq 0 \\ \text &\text x=y=z = 0 \\ \end \\ \varphi &= \sgn(y)\arccos\frac = \begin \arctan(\frac) &\text x > 0, \\ \arctan(\frac) + \pi &\text x < 0 \text y \geq 0, \\ \arctan(\frac) - \pi &\text x < 0 \text y < 0, \\ +\frac &\text x = 0 \text y > 0, \\ -\frac &\text x = 0 \text y < 0, \\ \text &\text x = 0 \text y = 0. \end \end The
inverse tangent In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Sp ...
denoted in must be suitably defined, taking into account the correct quadrant of . See the article on atan2. Alternatively, the conversion can be considered as two sequential rectangular to polar conversions: the first in the Cartesian plane from to , where is the projection of onto the -plane, and the second in the Cartesian -plane from to . The correct quadrants for and 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 plane, that is inclination from the direction, and that the azimuth angles are measured from the Cartesian axis (so that the axis has ). If ''θ'' measures elevation from the reference plane instead of inclination from the zenith the arccos above becomes an arcsin, and the and below become switched. Conversely, the Cartesian coordinates may be retrieved from the spherical coordinates (''radius'' , ''inclination'' , ''azimuth'' ), where , , , by : \begin x &= r \sin\theta \, \cos\varphi, \\ y &= r \sin\theta \, \sin\varphi, \\ z &= r \cos\theta. \end


Cylindrical coordinates

Cylindrical coordinates 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 d ...
(''axial'' ''radius'' ''ρ'', ''azimuth'' ''φ'', ''elevation'' ''z'') may be converted into spherical coordinates (''central radius'' ''r'', ''inclination'' ''θ'', ''azimuth'' ''φ''), by the formulas : \begin r &= \sqrt, \\ \theta &= \arctan\frac = \arccos\frac, \\ \varphi &= \varphi. \end Conversely, the spherical coordinates may be converted into cylindrical coordinates by the formulae : \begin \rho &= r \sin \theta, \\ \varphi &= \varphi, \\ z &= r \cos \theta. \end These formulae assume that the two systems have the same origin and same reference plane, measure the azimuth angle in the same senses from the same axis, and that the spherical angle is inclination from the cylindrical 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 : ax^2 + by^2 + cz^2 = d. The modified spherical coordinates of a point in P in the ISO convention (i.e. for physics: ''radius'' , ''inclination'' , ''azimuth'' ) can be obtained from its Cartesian coordinates by the formulae : \begin x &= \frac r \sin\theta \, \cos\varphi, \\ y &= \frac r \sin\theta \, \sin\varphi, \\ z &= \frac r \cos\theta, \\ r^ &= ax^2 + by^2 + cz^2. \end An infinitesimal volume element is given by : \mathrmV = \left, \frac\ \, dr\,d\theta\,d\varphi = \frac r^2 \sin \theta \,\mathrmr \,\mathrm\theta \,\mathrm\varphi = \frac r^2 \,\mathrmr \,\mathrm\Omega. The square-root factor comes from the property of the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
that allows a constant to be pulled out from a column: : \begin ka & b & c \\ kd & e & f \\ kg & h & i \end = k \begin a & b & c \\ d & e & f \\ g & h & i \end.


Integration and differentiation in spherical coordinates

The following equations (Iyanaga 1977) assume that the colatitude is the inclination from the (polar) axis (ambiguous since , , and are mutually normal), as in the ''physics convention'' discussed. The
line element In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc ...
for an infinitesimal displacement from to is \mathrm\mathbf = \mathrmr\,\hat + r\,\mathrm\theta \,\hat + r \sin \, \mathrm\varphi\,\mathbf, where \begin \hat &= \sin \theta \cos \varphi \,\hat + \sin \theta \sin \varphi \,\hat + \cos \theta \,\hat, \\ \hat &= \cos \theta \cos \varphi \,\hat + \cos \theta \sin \varphi \,\hat - \sin \theta \,\hat, \\ \hat &= - \sin \varphi \,\hat + \cos \varphi \,\hat \end are the local orthogonal
unit vectors In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vec ...
in the directions of increasing , , and , respectively, and , , and are the unit vectors in Cartesian coordinates. The linear transformation to this right-handed coordinate triplet is a
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \en ...
, R = \begin \sin\theta\cos\varphi&\sin\theta\sin\varphi& \cos\theta\\ \cos\theta\cos\varphi&\cos\theta\sin\varphi&-\sin\theta\\ -\sin\varphi&\cos\varphi &0 \end. 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 In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity ma ...
, that is, its inverse is simply its
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
. The Cartesian unit vectors are thus related to the spherical unit vectors by: \begin\mathbf \\ \mathbf \\ \mathbf \end = \begin \sin\theta\cos\varphi & \cos\theta\cos\varphi & -\sin\varphi \\ \sin\theta\sin\varphi & \cos\theta\sin\varphi & \cos\varphi \\ \cos\theta & -\sin\theta & 0 \end \begin \boldsymbol \\ \boldsymbol \\ \boldsymbol \end The general form of the formula to prove the differential line element, is \mathrm\mathbf = \sum_i \frac \,\mathrmx_i = \sum_i \left, \frac\ \frac \, \mathrmx_i = \sum_i \left, \frac\ \,\mathrmx_i \, \hat_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 = \begin r \sin\theta \, \cos\varphi \\ r \sin\theta \, \sin\varphi \\ r \cos\theta \end. Thus, \frac = \begin \sin\theta \, \cos\varphi \\ \sin\theta \, \sin\varphi \\ \cos\theta \end=\mathbf, \quad \frac = \begin r \cos\theta \, \cos\varphi \\ r \cos\theta \, \sin\varphi \\ -r \sin\theta \end=r\,\hat, \quad \frac = \begin -r \sin\theta \, \sin\varphi \\ r \sin\theta \, \cos\varphi \\ 0 \end = r \sin\theta\,\mathbf . The desired coefficients are the magnitudes of these vectors: \left, \frac\ = 1, \quad \left, \frac\ = r, \quad \left, \frac\ = r \sin\theta. The surface element spanning from to and to on a spherical surface at (constant) radius is then \mathrmS_r = \left\, \frac \times \frac\right\, \mathrm\theta \,\mathrm\varphi = \left, r \times r \sin \theta \= r^2 \sin\theta \,\mathrm\theta \,\mathrm\varphi ~. Thus the differential
solid angle In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The poi ...
is \mathrm\Omega = \frac = \sin\theta \,\mathrm\theta \,\mathrm\varphi. The surface element in a surface of polar angle constant (a cone with vertex the origin) is \mathrmS_\theta = r \sin\theta \,\mathrm\varphi \,\mathrmr. The surface element in a surface of azimuth constant (a vertical half-plane) is \mathrmS_\varphi = r \,\mathrmr \,\mathrm\theta. The
volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form :dV ...
spanning from to , to , and to is specified by the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as ...
of
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
s, J =\frac =\begin \sin\theta\cos\varphi&r\cos\theta\cos\varphi&-r\sin\theta\sin\varphi\\ \sin\theta \sin\varphi&r\cos\theta\sin\varphi&r\sin\theta\cos\varphi\\ \cos\theta&-r\sin\theta&0 \end, namely \mathrmV = \left, \frac\ \,\mathrmr \,\mathrm\theta \,\mathrm\varphi= r^2 \sin\theta \,\mathrmr \,\mathrm\theta \,\mathrm\varphi = r^2 \,\mathrmr \,\mathrm\Omega ~. Thus, for example, a function can be integrated over every point in by the
triple integral In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-numbe ...
\int\limits_0^ \int\limits_0^\pi \int\limits_0^\infty f(r, \theta, \varphi) r^2 \sin\theta \,\mathrmr \,\mathrm\theta \,\mathrm\varphi ~. The
del Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes ...
operator in this system leads to the following expressions for the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
,
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
,
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fi ...
and (scalar)
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
, \begin \nabla f = &\hat + \hat + \hat, \\ pt\nabla\cdot \mathbf = & \frac\left( r^2 A_r \right) + \frac \left( \sin\theta A_\theta \right) + \frac , \\ pt\nabla \times \mathbf = & \frac\left( \left( A_\varphi\sin\theta \right) - \right) \hat \\ pt& + \frac 1 r \left( - \left( r A_\varphi \right) \right) \hat \\ pt& + \frac 1 r \left( \left( r A_\theta \right) - \right) \hat, \\ pt\nabla^2 f = & \left(r^2 \right) + \left(\sin\theta \right) + \\ pt= & \left(\frac + \frac \frac\right)f + \left(\sin\theta \frac\right)f + \frac\fracf ~. \end Further, the inverse Jacobian in Cartesian coordinates is J^ = \begin \dfrac&\dfrac&\dfrac\\\\ \dfrac&\dfrac&\dfrac\\\\ \dfrac&\dfrac&0 \end. The metric tensor in the spherical coordinate system is g = J^T J .


Distance in spherical coordinates

In spherical coordinates, given two points with being the azimuthal coordinate :\begin &= (r,\theta,\varphi), \\ &= (r',\theta',\varphi') \end The distance between the two points can be expressed as :\begin &= \sqrt \end


Kinematics

In spherical coordinates, the position of a point or particle (although better written as a
triple Triple is used in several contexts to mean "threefold" or a " treble": Sports * Triple (baseball), a three-base hit * A basketball three-point field goal * A figure skating jump with three rotations * In bowling terms, three strikes in a row * ...
(r,\theta, \varphi)) can be written as : \mathbf = r \mathbf . Its velocity is then : \mathbf = \frac = \dot \mathbf + r\,\dot\theta\,\hat + r\,\dot\varphi \sin\theta\,\mathbf and its acceleration is : \begin \mathbf = \frac = & \left( \ddot - r\,\dot\theta^2 - r\,\dot\varphi^2\sin^2\theta \right)\mathbf \\ & + \left( r\,\ddot\theta + 2\dot\,\dot\theta - r\,\dot\varphi^2\sin\theta\cos\theta \right) \hat \\ & + \left( r\ddot\varphi\,\sin\theta + 2\dot\,\dot\varphi\,\sin\theta + 2 r\,\dot\theta\,\dot\varphi\,\cos\theta \right) \hat \end The angular momentum is : \mathbf = \mathbf \times \mathbf = \mathbf \times m\mathbf = m r^2 (- \dot\varphi \sin\theta\,\mathbf + \dot\theta\,\hat) Where m is mass. In the case of a constant or else , this reduces to vector calculus in polar coordinates. The corresponding angular momentum operator then follows from the phase-space reformulation of the above, : \mathbf= -i\hbar ~\mathbf \times \nabla =i \hbar \left(\frac \frac - \hat \frac\right). The torque is given as : \mathbf = \frac = \mathbf \times \mathbf = -m \left(2r\dot\dot\sin\theta + r^2\ddot\sin + 2r^2\dot\dot\cos \right)\hat + m \left(r^2\ddot + 2r\dot\dot - r^2\dot^2\sin\theta\cos\theta \right) \hat The kinetic energy is given as : E_k = \fracm \left \left(\dot^2\right) + \left(r\dot\right)^2 + \left(r\dot\sin\theta\right)^2 \right


See also

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Notes


Bibliography

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External links

*
MathWorld description of spherical coordinates


{{Orthogonal coordinate systems Three-dimensional coordinate systems Orthogonal coordinate systems fi:Koordinaatisto#Pallokoordinaatisto