Spherical coordinate system

Definition and coordinate relations

The standard conversion between spherical coordinates (r, θ, φ) and Cartesian coordinates (x, y, z) is given by the relations:

• x = r sin θ cos φ
• y = r sin θ sin φ
• z = r cos θ

Here r ≥ 0. Typical ranges are 0 ≤ θ ≤ π for the polar angle and 0 ≤ φ < 2π for the azimuth. If a point lies on a sphere of known radius, only the two angles are needed to specify its direction on that sphere; see sphere.

Common conventions and practical notes

Different disciplines use slightly different conventions for which angle is called θ or φ and for their zero references. In mathematical physics it is common to take θ as the polar angle from the positive z-axis and φ as the azimuth measured from the positive x-axis in the x-y plane. In some geodetic and navigation contexts the latitude-like angle is measured from the equator rather than from the pole, so the relationship to geographic latitude and longitude must be checked: geographic longitude corresponds to the azimuthal coordinate, while geographic latitude is related to the polar coordinate by latitude = 90° − θ when θ is colatitude. For Earth-related uses see longitude, latitude, and general remarks about Earth.

Differential elements, unit vectors, and singularities

The spherical system introduces non-constant unit vectors and a position-dependent Jacobian. Important formulas for calculus in these coordinates include the line element ds^2 = dr^2 + r^2 dθ^2 + r^2 sin^2 θ dφ^2 and the volume element dV = r^2 sin θ dr dθ dφ. Unit basis vectors are commonly labeled r̂, θ̂ and φ̂; unlike Cartesian unit vectors, they change direction with position. Two practical singularities occur where sin θ = 0 (the polar axis) causing the azimuth φ to be undefined, and along any choice of branch cut for φ when it wraps from 2π back to 0.

Applications, examples and conversions

Spherical coordinates are widely used in problems with radial symmetry: gravitational and electric fields around a point mass or charge, wave equations in spheres, antenna patterns, celestial coordinates for stars, and in geographic and oceanographic mapping. Converting from spherical to Cartesian is often performed when applying numerical methods or plotting: for instance, the point with coordinates r = 2, θ = π/2, φ = 0 corresponds to Cartesian (x, y, z) = (2, 0, 0).

Because conventions vary, always state the definitions of θ and φ when giving coordinates. When integrating over a sphere remember the factor sin θ in the volume or surface integrals. For visual introductions and figures consult the schematic images and references provided by textbooks and online resources: angles, distance, and general sphere references at sphere.

Historical methods for locating points on the globe and the development of spherical trigonometry predate modern notation; the compact three-parameter spherical coordinate representation became standard with the rise of analytic geometry and vector calculus. Today, clear specification of conventions and careful treatment of singular directions are the main practical concerns when using spherical coordinates in science and engineering.