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:
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.
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.
Conversely, Cartesian coordinates may be retrieved from spherical coordinates by:
Latitude is the complement of the zenith or colatitude, and can be converted by:
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:
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.
In spherical coordinates the position of a point is written,
Publication No. WO/2010/075754 Published on July 8, Assigned to ZTE for Radiation Performance Testing Method, System (Chinese Inventor)
Jul 08, 2010; GENEVA, July 10 -- Zhong Yu, China, has developed a radiation performance testing method and system. The patent has been assigned...