spherical coordinates

Spherical coordinatesMathworldPlanetmath are a system of coordinates for 3, or more generally n. One coordinate is the distanceMathworldPlanetmath from the origin, which can be thought of as the radius of the sphere centred at the origin on which the point lies. The other coordinates are angles that specify the position of the point on this sphere.

In 3 the coordinates are given by

(xyz) =(rsinϕcosθrsinϕsinθrcosϕ),

where r is the distance from the origin, θ is the azimuthal angle defined for θ[0,2π), and ϕ[0,π] is the polar angle. Note that ϕ=0 corresponds to the top of the sphere and ϕ=π corresponds to the bottom of the sphere. There is a clash between the mathematicians’ and the physicists’ definition of spherical coordinates, interchanging both the direction of ϕ and the choice of names for the two angles (physicists often use θ as the azimuthal angle and ϕ as the polar one).

Spherical coordinates are a generalizationPlanetmathPlanetmath of polar coordinates, and can be further generalized to n, with n-2 polar angles ϕ1,,ϕn-2 and one azimuthal angle θ:

(x1x2xkxn-1xn) =(rcosϕ1rsinϕ1cosϕ2r(i=1k-1sinϕi)cosϕkrsinϕ1sinϕ2cosθrsinϕ1sinϕ2sinϕn-2sinθ.).

These are sometimes called hyperspherical coordinates if n>3.

Title spherical coordinates
Canonical name SphericalCoordinates
Date of creation 2013-05-08 10:16:57
Last modified on 2013-05-08 10:16:57
Owner yark (2760)
Last modified by unlord (1)
Numerical id 17
Author yark (1)
Entry type Definition
Classification msc 51M05
Related topic Sphere
Related topic CylindricalCoordinates
Defines hyperspherical coordinates
Defines azimuthal angle
Defines polar angle