Spherical coordinate system

A spherical coordinate system, with origin O and azimuth axis A. The point has radius r = 4, elevation θ = 50°, and azimuth φ = 130°.

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance from a fixed origin, the elevation angle of that point from a fixed plane, and the azimuth angle of its orthogonal projection on that plane, from a fixed direction on the same. The elevation angle is often replaced by the inclination angle measured from the zenith, the direction perpendicular to 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 and astronomy, the elevation and azimuth (or quantities very close to them) are called the latitude and longitude, respectively; and the radial distance is replaced by an altitude (measured from a central point or from a sea level surface).

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

An alternate spherical coordinate system, using inclination from the zenith direction Z instead of elevation. The point has radius r = 4, inclination θ = 70°, and azimuth φ = 130°.

Illustration of spherical coordinates. The red sphere shows the points with r = 2, the blue cone shows the points with inclination (or elevation) θ = 45°, and the yellow half-plane shows the points with azimuth φ = −60°. The zenith direction is vertical, and the zero-azimuth axis is highlighted in green. The spherical coordinates (2,45°,−60°) determine the point of space where those three surfaces intersect, shown as a black sphere.

The three coordinates (r, θ, φ) of a point P are defined as:

• the radial distance r is the Euclidean distance from the origin to the point P.
• the inclination θ is the angle between the zenith direction and the line formed between the origin and P.
• the azimuth φ is the angle between the reference direction on the chosen plane and the line from the origin to the projection of P on the plane.

If the inclination θ is zero or 180°, the azimuth φ is indeterminate. If the radius r is zero, θ and φ are both indeterminate.

The elevation angle is 90° minus the inclination angle.

To plot a point from its spherical coordinates, go r units from the origin along the positive z-axis, rotate θ about the y-axis in the direction of the positive x-axis and rotate φ about the z-axis in the direction of the positive y-axis.

Conventions

Several different conventions exist for representing the three coordinates. 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 33-11.

However, some authors (including many American mathematicians) use φ for inclination (or elevation) and θ for azimuth. Some authors may also list the azimuth before the inclination (or elevation), and/or use ϝ 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 coordinate system and the three-dimensional cylindrical coordinate system, where θ is often used for the azimuth.

The angles are typically measured in degrees (°) or radians (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.

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+180°) for any r, θ, and φ. Moreover, (r, −θ, φ) is equivalent to(r, θ, φ+180°). Finally, if θ is zero or 180° (or, in general n×180°, for any integer n), then the azimuth angle is arbitrary; and if r is zero, both azimuth and elevation are arbitrary. Analogous (but different) equivalencies hold when one uses elevation instead of inclination.

If it is necessary to define a unique set of spherical coordinates for each point has, one may restrict their ranges. A common choice is

$r \geq 0\,$

$0^\circ \leq \theta \leq 180^\circ \left(\pi\!\ rad \right)\,$

$0^\circ \leq \varphi \leq 360^\circ \left(2\pi\!\ rad \right)\,$

However, the azimuth φ is often restricted to the interval (−180°, +180°], or (−π, +π] in radians, instead of [0, 360°). This is the standard convention for geographic longitude.

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

Applications

The geographic coordinate system uses the azimuth and elevation of the spherical coordinate system to express locations on Earth, calling them latitude and longitude. Just as the two-dimensional Cartesian coordinate system 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 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 and the Helmholtz equation, allow a separation of variables in spherical coordinates. The angular portions of the solutions to such equations take the form of spherical harmonics.

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.

The output pattern of an industrial loudspeaker shown using spherical polar plots taken at six frequencies

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

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 can be obtained from its Cartesian coordinates by the formulas

$r=\sqrt{x^2 + y^2 + z^2}$

${\varphi}=\operatorname{atan2}(y,x)$

${\theta}=\arccos \left( {\frac{z}{\sqrt{x^2 + y^2 + z^2}}} \right)$

where atan2(y,x) is a variant of the arctangent function that returns the angle from the x-axis to the vector (x,y) in the full range $(-\pi,\pi]$. (One cannot use the ordinary arctangent function, ${\varphi}=\arctan(y/x)$, because it returns the same angle for (x,y) and (−x,−y)).

These formulas assume that the two systems have the same origin, that the spherical reference plane is the Cartesian x–y plane, and that the azimuth angles are measured from the Cartesian x axis, so that the y axis has φ=+90°.

Conversely, the Cartesian coordinates may be retrieved from spherical coordinates by

${x}=r \, \cos\varphi \, \sin\theta \quad$

${y}=r \, \sin\varphi \, \sin\theta \quad$

${z}=r \, \cos\theta \quad$

Geographic coordinates

To a first approximation, the geographic coordinate system uses elevation angle (latitude), usually denoted by δ or θ, in degrees north of the equator 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 (most commonly that of the Greenwich Observatory), so its domain is −180° ≤ φ ≤ 180° For positions on the Earth or other solid celestial body, the reference plane is usually taken to be the plane perpendicular to the axis of rotation; and altitude above some conventional sea level or "mean" surface level is used instead of radial distance. In astronomy one may measure latitude either from the celestial equator (defined by the Earth's rotation) or the plane of the ecliptic (defined by Earth's orbit around the sun; or, sometimes, the galactic equator (defined by the rotation or the galaxy).

Latitude δ is the complement of the zenith angle θ or colatitude, and can be converted by:

${\delta}=90^\circ - \theta$, or

${\theta}=90^\circ - \delta$,

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, the geographical coordinate system is quite complex, and the positions implied by these simple formulas may be wrong by several kilometers. The precise standard meanings of latitude, longitude and altitude are currently defined by the World Geodetic System (WGS), and take into account the flattening of the Earth at the poles (about 21 km) and many other details.

Cylindrical coordinates

Cylindrical coordinates may be converted into spherical coordinates by:

$r=\sqrt{\rho^2+z^2}$

${\theta}=\operatorname{arctan}(\rho,z)$

${\varphi}=\varphi \quad$

Spherical coordinates may be converted into cylindrical coordinates by:

$\rho = r \sin \theta \,$

$\varphi = \varphi \,$

$z = r \cos \theta \,$

Integration and differentiation in spherical coordinates

The following equations are in the convention where $\theta$ is inclination from the normal axis:

The line element for an infinitesimal displacement from $(r,\theta,\varphi)$ to $(r+\mathrm{d}r, \,\theta+\mathrm{d}\theta, \, \varphi+\mathrm{d}\varphi)$ is

$\mathrm{d}\mathbf{r} = \mathrm{d}r\,\boldsymbol{\hat r} + r\,\mathrm{d}\theta \,\boldsymbol{\hat\theta } + r \sin{\theta} \mathrm{d}\varphi\,\mathbf{\boldsymbol{\hat \varphi}}.$

where $\boldsymbol{\hat r},\boldsymbol{\hat\theta },\boldsymbol{\hat \varphi}$ are the local orthogonal unit vectors in the directions of increasing $r,\theta,\varphi$, respectively.

The surface element spanning from $\theta$ to $\theta+\mathrm{d}\theta$ and $\varphi$ to $\varphi+\mathrm{d}\varphi$ on a spherical surface at (constant) radius $r$ is

$\mathrm{d}S=r^2\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\varphi$

The volume element spanning from $r$ to $r+\mathrm{d}r$, $\theta$ to $\theta+\mathrm{d}\theta$ and $\varphi$ to $\varphi+\mathrm{d}\varphi$ is

$\mathrm{d}V=r^2\sin\theta\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\varphi$

The del operator in this system is written as

$\nabla = \boldsymbol{\hat r}\frac{\partial}{\partial r} + \boldsymbol{\hat \theta}\frac{1}{r}\frac{\partial}{\partial \theta} + \boldsymbol{\hat \varphi}\frac{1}{r \sin\theta}\frac{\partial}{\partial \varphi}.$

Kinematics

In spherical coordinates the position of a point is written,

$\mathbf{r} = r \mathbf{e}_r$

its velocity is then,

$\mathbf{v} = \dot{r} \mathbf{e}_r + r\,\dot\theta\,\mathbf{e}_\theta + r\,\dot\varphi\,\sin\theta \mathbf{e}_\varphi$

and its acceleration is,

$\mathbf{a} = \left( \ddot{r} - r\,\dot\theta^2 - r\,\dot\varphi^2\sin^2\theta \right)\mathbf{e}_r$

$+ \left( r\,\ddot\theta + 2\dot{r}\,\dot\theta - r\,\dot\varphi^2\sin\theta\cos\theta \right) \mathbf{e}_\theta$

$+ \left( r\ddot\varphi\,\sin\theta + 2\dot{r}\,\dot\varphi\,\sin\theta + 2 r\,\dot\theta\,\dot\varphi\,\cos\theta \right) \mathbf{e}_\varphi$

See also

• Vector fields in cylindrical and spherical coordinates
• Del in cylindrical and spherical coordinates
• List of canonical coordinate transformations
• Sphere
• Hypersphere

Bibliography

• Philip M. Morse, Herman Feshbach (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. 658. ISBN 0-07-043316-X,. 

• Henry Margenau, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 177–178. 

• Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 174–175. 

• Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. 95–96. 

• 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. ed.). New York: Springer-Verlag. 24–27 (Table 1.05). ISBN 978-0387184302. 

External links

• MathWorld description of spherical coordinates
• Coordinate Converter - converts between polar, Cartisian and spherical coordinates