Spherical CoordinatesEdit

Spherical coordinates offer a way to describe points in three-dimensional space using a radial distance and two angles. This system is especially natural when dealing with problems that exhibit rotational symmetry around an axis or around a central point, such as gravitational fields, planetary orbits, or lighting in computer graphics. While the more ubiquitous Cartesian coordinates (the standard x, y, z system) are convenient for many algebraic manipulations, spherical coordinates often simplify problems with spherical or radial geometry and connect neatly to the geometry of spheres and cones. For readers exploring geometry, physics, or engineering, they provide a complementary perspective to Cartesian coordinates and Cylindrical coordinates.

In three-dimensional space, a point is represented by a triplet (r, θ, φ) in one common convention, where r is the radial distance from the origin, θ is an angular measure in the horizontal plane, and φ is a polar angle measuring tilt from the positive z-axis. It is important to note that different communities use slightly different letter assignments for these angles, which can affect conversions between coordinate systems. In some mathematical traditions, the roles of θ and φ are swapped relative to the physics convention; both conventions are widely used and are interchangeable once one keeps track of the definitions. See the entries on coordinate system theory for a broader discussion of these choices.

Definition and conventions

The coordinate triplet and its range

r: nonnegative real number, r ≥ 0, giving the distance from the origin.
θ: azimuthal angle in the xy-plane, typically measured from the positive x-axis toward the positive y-axis, with 0 ≤ θ < 2π.
φ: polar (colatitudinal) angle from the positive z-axis, with 0 ≤ φ ≤ π.

These ranges define one standard convention, sometimes called the physics convention. A second, frequently used convention in mathematics swaps the roles of φ and θ, resulting in - θ: the polar angle from the positive z-axis, 0 ≤ θ ≤ π - φ: the azimuth in the xy-plane, 0 ≤ φ < 2π

The two conventions are equivalent once one keeps explicit track of which angle plays which role. The geometry of the coordinate surfaces is transparent: concentric spheres r = constant, half-planes or curved surfaces given by φ = constant (cones), and vertical planes given by θ = constant (radial half-planes).

Conversions to and from Cartesian coordinates

From the physics convention (r, θ, φ): - x = r sinφ cosθ - y = r sinφ sinθ - z = r cosφ

From the math convention (r, θ, φ) where θ is the polar angle and φ the azimuth: - x = r cosφ sinθ - y = r sinφ sinθ - z = r cosθ

Conversely, from Cartesian coordinates (x, y, z) to the physics convention: - r = sqrt(x^2 + y^2 + z^2) - θ = atan2(y, x) - φ = arccos(z / r)

And for the math convention, θ and φ are swapped in the inverse relations.

Differential elements and the metric

The Jacobian determinant governing volume elements shows how “volume” scales when switching to spherical coordinates. In the common physics convention, the volume element is: - dV = r^2 sinφ dr dφ dθ

Surface elements on a sphere of radius r are: - dS = r^2 sinφ dφ dθ

These expressions reflect the geometry of spheres and cones in spherical coordinates and are essential in applications involving integration over regions with spherical symmetry.

Geometric interpretation and relations to other systems

Surfaces r = constant are spheres centered at the origin.
Surfaces φ = constant are cones with apex at the origin, opening angle determined by φ.
Surfaces θ = constant are half-planes through the z-axis, rotated by θ about the z-axis.

Spherical coordinates are closely related to alternative coordinate systems: - Cylindrical coordinates (r, φ, z) separate horizontal radial distance from vertical height, combining circular symmetry in the xy-plane with vertical variation. - Polar coordinates describe points in a plane with a radial distance and an azimuthal angle; spherical coordinates extend this idea to three dimensions with an additional radial and polar component.

Calculations, physics, and applications

Physics and engineering

Spherical coordinates simplify the description of central forces, gravitational and electric fields, and wave problems with spherical symmetry. The Laplacian and other differential operators take forms in spherical coordinates that reveal radial, angular, and polar dependencies more naturally than in Cartesian coordinates. In electromagnetism and quantum mechanics, many problems reduce to solving equations in spherical coordinates due to boundary conditions aligned with spheres and cones. See discussions of Laplacian in different coordinate systems and applications to harmonic functions.

Astronomy and geoscience

Celestial mechanics and stellar structure often exploit spherical symmetry, especially in modeling radiative transfer and spherical shells in stars. In geodesy and planetary science, angular coordinates on spheres underpin common representations of celestial coordinates and planetary surfaces.

Computer graphics and visualization

Lighting models, environment maps, and volumetric rendering frequently use spherical coordinates to describe directions, radiance, or sampling patterns on shells around objects. Transformations between coordinate systems are routine in graphics pipelines, with clear relationships to vector representations and matrix transformations.

History and development

The idea of expressing points by a radial distance and angular directions emerges from early advances in analytic geometry and astronomy, where spherical symmetry naturally arises in the study of the heavens and celestial spheres. Over the centuries, mathematicians and physicists formalized the coordinate systems, their conversion formulas, and their differential properties, enabling a wide range of analytic and computational techniques in science and engineering. The dual conventions for the angles reflect differing historical emphases on geometry vs. applied physics, but the underlying geometry remains the same.