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 zaxis to the point, and the azimuth angle from the positive xaxis to the orthogonal projection of the point in the xy plane.
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[edit] Notation
Several different conventions exist for representing the three coordinates. In accordance with the International Organisation for Standardisation (ISO 3111), 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 lefthanded 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 twodimensional polar coordinate system and the threedimensional 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.^{[1]} 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.
[edit] Definition
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 zaxis and the line formed between the origin and P.
 0 ≤ φ < 2π is the angle between the positive xaxis and the line from the origin to the P projected onto the xyplane.
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 zaxis, rotate θ about the yaxis in the direction of the positive xaxis and rotate φ about the zaxis in the direction of the positive yaxis.
[edit] Coordinate system conversions
As the spherical coordinate system is only one of many threedimensional coordinate systems, there exist equations for converting coordinates between the spherical coordinate system and others.
[edit] 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:
[edit] 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 xyplane with a domain 90° ≤ θ ≤ 90°. The longitude is measured in degrees east or west from 0°, so its domain is 180° ≤ φ ≤ 180°.
[edit] Cylindrical coordinate system
 Further information: Cylindrical coordinate system
The cylindrical coordinate system is a threedimensional extrusion of the polar coordinate system, with an z coordinate to describe a point's height above or below the xyplane. 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:
[edit] Applications
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 twodimensional Cartesian coordinate system is useful on the plane, a twodimensional 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 x^{2} + y^{2} + z^{2} = c^{2} 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
[edit] Kinematics
In spherical coordinates the position of a point is written,
its velocity is then,
and its acceleration is,
[edit] Notes
[edit] See also
 Vector fields in cylindrical and spherical coordinates
 Del in cylindrical and spherical coordinates
 List of canonical coordinate transformations
 Sphere
 Hypersphere
 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
 Flatring cyclide coordinates
 Flatdisk cyclide coordinates
 Bicyclide coordinates
 Capcyclide coordinates
[edit] Bibliography
 Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGrawHill. pp. 658. ISBN 007043316X, LCCN 5211515.
 Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 177–178. LCCN 5510911.
 Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGrawHill. pp. 174–175. LCCN 5914456, ASIN B0000CKZX7.
 Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. pp. 95–96. LCCN 6725285.
 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: SpringerVerlag. pp. 24–27 (Table 1.05). ISBN 9780387184302.
[edit] External links
 MathWorld description of spherical coordinates
 Coordinate Converter  converts between polar, Cartisian and spherical coordinates
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