- Spherical coordinate system
mathematics, the spherical coordinate system is a coordinate systemfor representing geometric figures in three dimensions using three coordinates: the radial distance of a point from a fixed origin, the zenithangle from the positive z-axis to the point, and the azimuthangle from the positive x-axis to the orthogonal projectionof 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 systemand 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. [cite web
url = http://mathworld.wolfram.com/SphericalCoordinates.html
title = Spherical Coordinates
Eric W. Weisstein
accessdate = 2007-04-10] 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
The three spherical coordinates are obtained from
Cartesian coordinatesby::::Note that the arctangent must be defined suitably so as to take account of the correct quadrant of . The atan2or equivalent function accomplishes this for computational purposes.
Conversely, Cartesian coordinates may be retrieved from spherical coordinates by::::
Geographic coordinate system
The geographic coordinate system is an alternate version of the spherical coordinate system, used primarily in
geographythough also in mathematics and physicsapplications. 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
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::::
geographic coordinate systemapplies the two angles of the spherical coordinate system to express locations on Earth, calling them latitudeand longitude. Just as the two-dimensional Cartesian coordinate systemis 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 "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 equationand the Helmholtz equation, allow a separation of variablesin 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,:its velocity is then,:and its acceleration is,:::
Vector fields in cylindrical and spherical coordinates
Del in cylindrical and spherical coordinates
List of canonical coordinate transformations
* | pages = p. 658
* | pages = pp. 177–178
*, ASIN B0000CKZX7 | pages = pp. 174–175
* | pages = pp. 95–96
* [http://mathworld.wolfram.com/SphericalCoordinates.html MathWorld description of spherical coordinates]
Wikimedia Foundation. 2010.
Look at other dictionaries:
spherical coordinate system — sferinė koordinačių sistema statusas T sritis fizika atitikmenys: angl. spherical coordinate system vok. Kugelkoordinatensystem, n rus. сферическая система координат, f pranc. système de coordonnées sphérique, m … Fizikos terminų žodynas
spherical coordinate system — In geometry, a coordinate system in which any point in three dimensional space is specified by its angle with respect to a polar axis and angle of rotation with respect to a prime meridian on a sphere of a given radius. In spherical coordinates a … Universalium
Coordinate system — For geographical coordinates on Wikipedia, see Wikipedia:WikiProject Geographical coordinates. 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… … Wikipedia
coordinate system — Math. any method that uses numbers to represent a point, line, or the like. * * * Arrangement of reference lines or curves used to identify the location of points in space. In two dimensions, the most common system is the Cartesian (after René… … Universalium
spherical coordinate — noun : any of three coordinates in space two being obtained by constructing in a plane a polar coordinate system and the third being the angle between this plane and a fixed plane containing the polar axis … Useful english dictionary
spherical coordinate — noun Date: circa 1864 one of three coordinates that are used to locate a point in space and that comprise the radius of the sphere on which the point lies in a system of concentric spheres, the angle formed by the point, the center, and a given… … New Collegiate Dictionary
Polar coordinate system — Points in the polar coordinate system with pole O and polar axis L. In green, the point with radial coordinate 3 and angular coordinate 60 … Wikipedia
Cylindrical coordinate system — A cylindrical coordinate system with origin O, polar axis A, and longitudinal axis L. The dot is the point with radial distance ρ = 4, angular coordinate φ = 130°, and height z = 4. A cylindrical coordinate system is … Wikipedia
Geographic coordinate system — Map of Earth showing lines of latitude (horizontally) and longitude (vertically), Eckert VI projection; large version (pdf, 3.12MB) … Wikipedia
Cartesian coordinate system — Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2, 3) in green, (−3, 1) in red, (−1.5, −2.5) in blue, and the origin (0, 0) in purple. A Cartesian coordinate system specifies each point… … Wikipedia