In mathematics, the spherical coordinate system is a coordinate system for representing geometric figures in three dimensions using three coordinates: the radial distance of a point from a fixed origin, the zenith angle from the positive z-axis to the point, and the azimuth angle from the positive x-axis to the orthogonal projection of the point in the x-y plane.
Several different conventions exist for representing the three coordinates. In accordance with the International Organisation for Standardisation (ISO 31-11), in physics they are typically notated as (r, θ, φ) for radial distance, zenith, and azimuth, respectively.
In (American) mathematics, the notation for zenith and azimuth are reversed as φ is used to denote the zenith angle and θ is used to denote the azimuthal angle. A further complication is that some mathematics texts list the azimuth before the zenith, but this convention is left-handed and should be avoided. The mathematical convention has the advantage of being most compatible in the meaning of θ with the traditional notation for the two-dimensional polar coordinate system and the three-dimensional cylindrical coordinate system, while the "physics" convention has broader acceptance geographically. Some users of the "physics" convention also use φ for polar coordinates to avoid the first problem (as is the standard ISO for cylindrical coordinates). Other notation uses ρ for radial distance. The notation convention of the author of any work pertaining to spherical coordinates should always be checked before using the formulas and equations of that author. This article uses the standard physics convention.
The three coordinates (r, θ, φ) are defined as:
- r ≥ 0 is the distance from the origin to a given point P.
- 0 ≤ θ ≤ π is the angle between the positive z-axis and the line formed between the origin and P.
- 0 ≤ φ < 2π is the angle between the positive x-axis and the line from the origin to the P projected onto the xy-plane.
φ is referred to as the azimuth, while θ is referred to as the zenith, colatitude or polar angle.
θ and φ lose significance when r = 0 and φ loses significance when sin(θ) = 0 (at θ = 0 and θ = π).
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.
 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 coordinate system
- Further information: Cartesian coordinate system
The three spherical coordinates are obtained from Cartesian coordinates by:
where atan2(y,x) is a variant of the arctangent function that can return angles outside the range [ − π / 2,π / 2].
Conversely, Cartesian coordinates may be retrieved from spherical coordinates by:
 Geographic coordinate system
- Further information: Geographic coordinate system
The geographic coordinate system is an alternate version of the spherical coordinate system, used primarily in geography though also in mathematics and physics applications. In geography, ρ is usually dropped or replaced with a value representing elevation or altitude.
Latitude is the complement of the zenith or colatitude, and can be converted by:
- , or
though latitude is typically represented by θ as well. This represents a zenith angle originating from the xy-plane with a domain -90° ≤ θ ≤ 90°. The longitude is measured in degrees east or west from 0°, so its domain is -180° ≤ φ ≤ 180°.
 Cylindrical coordinate system
- Further information: Cylindrical coordinate system
The cylindrical coordinate system is a three-dimensional extrusion of the polar coordinate system, with an z coordinate to describe a point's height above or below the xy-plane. The full coordinate tuple is (ρ, φ, z).
Cylindrical coordinates may be converted into spherical coordinates by:
Spherical coordinates may be converted into cylindrical coordinates by:
The geographic coordinate system applies the two angles 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 are symmetrical about a point; a sphere that has the Cartesian equation x2 + y2 + z2 = c2 has the very simple equation r = c in spherical coordinates. An example is in solving a triple integral with a sphere as its domain.
The surface element for a spherical surface is
The volume element is
Spherical coordinates are the natural coordinates for describing and analyzing physical situations where there is spherical symmetry, such as the potential energy field surrounding a sphere (or point) with mass or charge. Two important partial differential equations, 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 concept of spherical coordinates can be extended to higher dimensional spaces and are then referred to as hyperspherical coordinates.
The del operator in this system is written as
In spherical coordinates the position of a point is written,
its velocity is then,
and its acceleration is,
 See also
- Vector fields in cylindrical and spherical coordinates
- Del in cylindrical and spherical coordinates
- List of canonical coordinate transformations
- Orthogonal coordinates
- Two dimensional orthogonal coordinate systems
- Three dimensional orthogonal coordinate systems
- Cartesian coordinate system
- Cylindrical coordinate system
- Spherical coordinate system
- Parabolic coordinate system
- Parabolic cylindrical coordinates
- Paraboloidal coordinates
- Oblate spheroidal coordinates
- Prolate spheroidal coordinates
- Ellipsoidal coordinates
- Elliptic cylindrical coordinates
- Toroidal coordinates
- Bispherical coordinates
- Bipolar cylindrical coordinates
- Conical coordinates
- Flat-ring cyclide coordinates
- Flat-disk cyclide coordinates
- Bi-cyclide coordinates
- Cap-cyclide coordinates
- Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. pp. 658. ISBN 0-07-043316-X, LCCN 52-11515.
- Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 177–178. LCCN 55-10911.
- Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. pp. 174–175. LCCN 59-14456, ASIN B0000CKZX7.
- Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. pp. 95–96. LCCN 67-25285.
- 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. pp. 24–27 (Table 1.05). ISBN 978-0387184302.
 External links
- MathWorld description of spherical coordinates
- Coordinate Converter - converts between polar, Cartisian and spherical coordinates