Spherical coordinate system

# Spherical coordinate system

Overview

In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a spherical coordinate system is a coordinate system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

for three-dimensional space
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its inclination angle measured from a fixed zenith
Zenith
The zenith is an imaginary point directly "above" a particular location, on the imaginary celestial sphere. "Above" means in the vertical direction opposite to the apparent gravitational force at that location. The opposite direction, i.e...

direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.
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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a spherical coordinate system is a coordinate system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

for three-dimensional space
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its inclination angle measured from a fixed zenith
Zenith
The zenith is an imaginary point directly "above" a particular location, on the imaginary celestial sphere. "Above" means in the vertical direction opposite to the apparent gravitational force at that location. The opposite direction, i.e...

direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. The inclination angle is often replaced by the elevation angle measured from the reference plane.

The radial distance is also called the radius or radial coordinate, and the inclination may be called colatitude, zenith angle, normal angle, or polar angle.

In geography
Geography
Geography is the science that studies the lands, features, inhabitants, and phenomena of Earth. A literal translation would be "to describe or write about the Earth". The first person to use the word "geography" was Eratosthenes...

and astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

, the elevation and azimuth (or quantities very close to them) are called the latitude and longitude for geography and "declination" and "right ascension" for astronomy, respectively; and the radial distance is usually replaced by an altitude (measured from a central point or from a sea level
Sea level
Mean sea level is a measure of the average height of the ocean's surface ; used as a standard in reckoning land elevation...

surface).

The concept of spherical coordinates can be extended to higher dimensional spaces and are then referred to as hyperspherical coordinates.

## 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 P are then defined as follows:
Euclidean distance
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...

from the origin O to P.
• the inclination (or polar angle) is the angle between the zenith direction and the line segment OP.
• the azimuth (or azimuthal angle) is the signed angle measured from the azimuth reference direction to the orthogonal projection of the line segment OP on the reference plane.

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 90 degrees (π/2 radians) minus the inclination angle.

If the inclination is zero or 180 degrees (π radians), the azimuth is arbitrary. If the radius is zero, both azimuth and inclination are arbitrary.

In linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

, the vector from the origin O to the point P 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, θ, φ) to denote, respectively, radial distance, inclination (or elevation), and azimuth, is common practice in physics, and is specified by ISO standard 31-11
ISO 31-11
ISO 31-11 was the part of international standard ISO 31 that defines mathematical signs and symbols for use in physical sciences and technology...

.

However, some authors (including mathematicians) use φ for inclination (or elevation) and θ for azimuth, which "provides a logical extension of the usual polar coordinates notation". Some authors may also list the azimuth before the inclination (or elevation), and/or use ρ instead of r for radial distance. Some combinations of these choices result in a left-handed coordinate system. The standard convention (r, θ, φ) conflicts with the usual notation for the two-dimensional polar coordinates
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 fixed point and an angle from a fixed direction....

, where θ is often used for the azimuth. It may also conflict with the notation used for three-dimensional cylindrical coordinates
Cylindrical coordinate system
A cylindrical coordinate system is a three-dimensional coordinate systemthat specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis...

.

The angles are typically measured in degrees
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...

Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

s (rad), where 360° = 2π 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
North is a noun, adjective, or adverb indicating direction or geography.North is one of the four cardinal directions or compass points. It is the opposite of south and is perpendicular to east and west.By convention, the top side of a map is north....

and positive azimuth (longitude) angles are measured eastwards from some prime meridian
Prime Meridian
The Prime Meridian is the meridian at which the longitude is defined to be 0°.The Prime Meridian and its opposite the 180th meridian , which the International Date Line generally follows, form a great circle that divides the Earth into the Eastern and Western Hemispheres.An international...

.
Major conventions
coordinates corresponding local geographical directions
right/left-handed
right
right
left
Note: easting (), northing (), upwardness (). Local azimuth angle would be measured, e.g., counter-clockwise from to in the case of .

### Unique coordinates

Any spherical coordinate triplet (r, θ, φ) 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, θ, φ) is equivalent to (r, θ+180°, φ) for any r, θ, and φ. Moreover, (r, −θ, φ) is equivalent to(r, θ, φ+180°).

If it is necessary to define a unique set of spherical coordinates for each point, one may restrict their ranges. A common choice is:
r ≥ 0
0° ≤ θ ≤ 180° (π rad)
0° ≤ φ < 360° (2π rad)

However, the azimuth φ is often restricted to the interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

, or in radians, instead of . This is the standard convention for geographic longitude.

The range [0°, 180°] for inclination is equivalent to [−90°, +90°] for elevation (latitude).

Even with these restrictions, if θ is zero or 180° (elevation is 90° or -90°) then the azimuth angle is arbitrary; and if r 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 point from its spherical coordinates (r, θ, φ), where θ is inclination, move r units from the origin in the zenith direction, rotate by θ about the origin towards the azimuth reference direction, and rotate by φ about the zenith in the proper direction.

## Applications

The geographic coordinate system
Geographic coordinate system
A geographic coordinate system is a coordinate system that enables every location on the Earth to be specified by a set of numbers. The coordinates are often chosen such that one of the numbers represent vertical position, and two or three of the numbers represent horizontal position...

uses the azimuth and elevation of the spherical coordinate system to express locations on Earth, calling them respectively longitude
Longitude
Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

and latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

. Just as the two-dimensional Cartesian coordinate system
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

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, 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
Multiple integral
The multiple integral is a type of definite integral extended to functions of more than one real variable, for example, ƒ or ƒ...

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 x2 + y2 + z2 = c2 has the simple equation r = c in spherical coordinates.

Two important partial differential equations that arise in many physical problems, Laplace's equation
Laplace's equation
In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...

and the Helmholtz equation
Helmholtz equation
The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation\nabla^2 A + k^2 A = 0where ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude.-Motivation and uses:...

, allow a separation of variables
Separation of variables
In mathematics, separation of variables 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 on a different side of the equation....

in spherical coordinates. The angular portions of the solutions to such equations take the form of spherical harmonics
Spherical harmonics
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...

.

Another application is ergonomic design, where r 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
Loudspeaker
A loudspeaker is an electroacoustic transducer that produces sound in response to an electrical audio signal input. Non-electrical loudspeakers were developed as accessories to telephone systems, but electronic amplification by vacuum tube made loudspeakers more generally useful...

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
Game development
Game development is the software development process by which a video game is developed. Development is undertaken by a game developer, which may range from a single person to a large business. Mainstream games are normally funded by a publisher and take several years to develop. Indie games can...

to rotate the camera around the player's position.

## 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 (ρ, φ, θ) of a point can be obtained from its Cartesian coordinates
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

(x, y, z) by the formulae

The inverse tangent denoted in φ = arctan(y/x) must be suitably defined, taking into account the correct quadrant of (x,y). See article atan2
Atan2
In trigonometry, the two-argument function atan2 is a variation of the arctangent function. For any real arguments and not both equal to zero, is the angle in radians between the positive -axis of a plane and the point given by the coordinates on it...

.

Alternatively, the conversion can be considered as two sequential rectangular to polar conversions: the first in the Cartesian xy plane from (x,y) to (R,φ), where R is the projection of r onto the xy plane, and the second in the Cartesian z-R plane from (z,R) to (r,θ). The correct quadrants for φ and 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 xy plane, that θ is inclination from the z direction, and that the azimuth angles are measured from the Cartesian x axis (so that the y axis has φ=+90°). If θ measures elevation from the reference plane instead of inclination from the zenith the arccos above becomes an arcsin, and the cos θ and sin θ below become switched.

Conversely, the Cartesian coordinates may be retrieved from the spherical coordinates (r, φ, θ), where , , , by:

### Geographic coordinates

To a first approximation, the geographic coordinate system
Geographic coordinate system
A geographic coordinate system is a coordinate system that enables every location on the Earth to be specified by a set of numbers. The coordinates are often chosen such that one of the numbers represent vertical position, and two or three of the numbers represent horizontal position...

uses elevation angle (latitude), usually denoted by δ or θ, in degrees north of the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

plane, in the range −90° ≤ δ ≤ 90°, instead of inclination. The azimuth angle (longitude) is measured in degrees east or west from some conventional reference meridian
Meridian (geography)
A meridian is an imaginary line on the Earth's surface from the North Pole to the South Pole that connects all locations along it with a given longitude. The position of a point along the meridian is given by its latitude. Each meridian is perpendicular to all circles of latitude...

(most commonly that of the Greenwich Observatory), so its domain is −180° ≤ φ ≤ 180°. For positions on the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

or other solid celestial body
Celestial Body
Celestial Body is a Croatian film directed by Lukas Nola. It was released in 2000....

, the reference plane is usually taken to be the plane perpendicular to the axis of rotation. In astronomy one may measure latitude either from the celestial equator
Celestial equator
The celestial equator is a great circle on the imaginary celestial sphere, in the same plane as the Earth's equator. In other words, it is a projection of the terrestrial equator out into space...

(defined by the Earth's rotation) or the plane of the ecliptic
Ecliptic
The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...

(defined by Earth's orbit around the sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

) or, sometimes, the galactic equator (defined by the rotation of the galaxy
Galaxy
A galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas and dust, and an important but poorly understood component tentatively dubbed dark matter. The word galaxy is derived from the Greek galaxias , literally "milky", a...

).

The zenith angle or inclination, which is 90° minus the latitude and ranges from 0 to 180°, is called colatitude
Colatitude
In spherical coordinates, colatitude is the complementary angle of the latitude, i.e. the difference between 90° and the latitude.-Astronomical use:The colatitude is useful in astronomy because it refers to the zenith distance of the celestial poles...

in geography.

Altitude
Altitude or height is defined based on the context in which it is used . As a general definition, altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The reference datum also often varies according to the context...

above some reference surface, which may be the sea level
Sea level
Mean sea level is a measure of the average height of the ocean's surface ; used as a standard in reckoning land elevation...

or "mean" surface level for planets without liquid oceans. The radial distance r can be computed from the altitude by adding the mean radius of the planet's reference surface, which is approximately 6,360±11 km for the Earth.

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
World Geodetic System
The World Geodetic System is a standard for use in cartography, geodesy, and navigation. It comprises a standard coordinate frame for the Earth, a standard spheroidal reference surface for raw altitude data, and a gravitational equipotential surface that defines the nominal sea level.The latest...

(WGS), and take into account the flattening of the Earth at the poles (about 21 km) and many other details.

### Cylindrical coordinates

Cylindrical coordinates
Cylindrical coordinate system
A cylindrical coordinate system is a three-dimensional coordinate systemthat specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis...

(r, φ, z) may be converted into spherical coordinates (ρ, θ, φ), by the formulas

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

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

## Integration and differentiation in spherical coordinates

The following equations assume that θ is inclination from the normal axis:

The line element
Line element
A line element ds in mathematics can most generally be thought of as the change in a position vector in an affine space expressing the change of the arc length. An easy way of visualizing this relationship is by parametrizing the given curve by Frenet–Serret formulas...

for an infinitesimal displacement from to is

where are the local orthogonal unit vectors in the directions of increasing , respectively.

The surface element
Surface element
Surface element may refer to*Volume form*Surfel in 3D computer graphics*Differential , an infinitesimal portion of a surface...

spanning from to and to on a spherical surface at (constant) radius is

Thus the differential solid angle
Solid angle
The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...

is

The surface element in a surface of polar angle constant (a cone with vertex the origin) is

The surface element in a surface of azimuth constant (a vertical half-plane) is

The volume element
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...

spanning from to , to , and to is

Therefore every point in R3 can be integrated by the triple integral

The del
Del
In vector calculus, del is a vector differential operator, usually represented by the nabla symbol \nabla . When applied to a function defined on a one-dimensional domain, it denotes its standard derivative as defined in calculus...

operator in this system is not defined, and so the gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

, divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

and curl must be defined explicitly:

## Kinematics

In spherical coordinates the position of a point is written,
its velocity is then,
and its acceleration is,

In the case of a constant φ this reduces to vector calculus in polar coordinates.

• Celestial coordinate system
Celestial coordinate system
In astronomy, a celestial coordinate system is a coordinate system for mapping positions on the celestial sphere.There are different celestial coordinate systems each using a system of spherical coordinates projected on the celestial sphere, in analogy to the geographic coordinate system used on...

• Del in cylindrical and spherical coordinates
Del in cylindrical and spherical coordinates
This is a list of some vector calculus formulae of general use in working with various curvilinear coordinate systems.- Note :* This page uses standard physics notation. For spherical coordinates, \theta is the angle between the z axis and the radius vector connecting the origin to the point in...

• Euler angles
Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3-dimensional Euclidean space three parameters are required...

• Gimbal lock
Gimbal lock
Gimbal lock is the loss of one degree of freedom in a three-dimensional space that occurs when the axes of two of the three gimbals are driven into a parallel configuration, "locking" the system into rotation in a degenerate two-dimensional space....

• Hypersphere
Hypersphere
In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any...

• Jacobian matrix and determinant
• List of canonical coordinate transformations
• Sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

• Spherical harmonic
Spherical Harmonic
Spherical Harmonic is a science fiction novel from the Saga of the Skolian Empire by Catherine Asaro. It tells the story of Dyhianna Selei , the Ruby Pharaoh of the Skolian Imperialate, as she strives to reform her government and reunite her family in the aftermath of a devastating interstellar...

• Theodolite
Theodolite
A theodolite is a precision instrument for measuring angles in the horizontal and vertical planes. Theodolites are mainly used for surveying applications, and have been adapted for specialized purposes in fields like metrology and rocket launch technology...

• Vector fields in cylindrical and spherical coordinates
Vector fields in cylindrical and spherical coordinates
* This page uses standard physics notation. For spherical coordinates, \theta is the angle between the z axis and the radius vector connecting the origin to the point in question. \phi is the angle between the projection of the radius vector onto the x-y plane and the x axis...

• Yaw, pitch and roll