Spherical coordinate system

Both the use of symbols and the naming order of tuple coordinates differ among the several sources and disciplines.

According to the conventions of geographical coordinate systems, positions are measured by latitude, longitude, and height (altitude).

The spherical coordinates of a point P then are defined as follows: The sign of the azimuth is determined by designating the rotation that is the positive sense of turning about the zenith.

denotes radial distance, the polar angle—"inclination", or as the alternative, "elevation"—and the azimuthal angle.

However, some authors (including mathematicians) use the symbol ρ (rho) for radius, or radial distance, φ for inclination (or elevation) and θ for azimuth—while others keep the use of r for the radius; all which "provides a logical extension of the usual polar coordinates notation".

When the system is used to designate physical three-space, it is customary to assign positive to azimuth angles measured in the counterclockwise sense from the reference direction on the reference plane—as seen from the "zenith" side of the plane.

This convention is used in particular for geographical coordinates, where the "zenith" direction is north and the positive azimuth (longitude) angles are measured eastwards from some prime meridian.

In the case of (U, S, E) the local azimuth angle would be measured counterclockwise from S to E. Any spherical coordinate triplet (or tuple)

On the reverse view, any single point has infinitely many equivalent spherical coordinates.

To plot any dot from its spherical coordinates (r, θ, φ), where θ is inclination, the user would: move r units from the origin in the zenith reference direction (z-axis); then rotate by the amount of the azimuth angle (φ) about the origin from the designated azimuth reference direction, (i.e., either the x– or y–axis, see Definition, above); and then rotate from the z-axis by the amount of the θ angle.

Spherical coordinates are also useful in analyzing systems that have some degree of symmetry about a point, including: volume integrals inside a sphere; the potential energy field surrounding a concentrated mass or charge; or global weather simulation in a planet's atmosphere.

Polar plots help to show that many loudspeakers tend toward omnidirectionality at lower frequencies.

An important application of spherical coordinates provides for the separation of variables in two partial differential equations—the Laplace and the Helmholtz equations—that arise in many physical problems.

The angular portions of the solutions to such equations take the form of spherical harmonics.

The spherical coordinate system is also commonly used in 3D game development to rotate the camera around the player's position[4] Instead of inclination, the geographic coordinate system uses elevation angle (or latitude), in the range (aka domain) −90° ≤ φ ≤ 90° and rotated north from the equator plane.

The polar angle (inclination), which is 90° minus the latitude and ranges from 0 to 180°, is called colatitude in geography.

The azimuth angle (or longitude) of a given position on Earth, commonly denoted by λ, is measured in degrees east or west from some conventional reference meridian (most commonly the IERS Reference Meridian); thus its domain (or range) is −180° ≤ λ ≤ 180° and a given reading is typically designated "East" or "West".

Instead of the radial distance r geographers commonly use altitude above or below some local reference surface (vertical datum), which, for example, may be the mean sea level.

When needed, the radial distance can be computed from the altitude by adding the radius of Earth, which is approximately 6,360 ± 11 km (3,952 ± 7 miles).

However, modern geographical coordinate systems are quite complex, and the positions implied by these simple formulae may be inaccurate by several kilometers.

The precise standard meanings of latitude, longitude and altitude are currently defined by the World Geodetic System (WGS), and take into account the flattening of the Earth at the poles (about 21 km or 13 miles) and many other details.

A series of astronomical coordinate systems are used to measure the elevation angle from several fundamental planes.

These reference planes include: the observer's horizon, the galactic equator (defined by the rotation of the Milky Way), the celestial equator (defined by Earth's rotation), the plane of the ecliptic (defined by Earth's orbit around the Sun), and the plane of the earth terminator (normal to the instantaneous direction to the Sun).

The inverse tangent denoted in φ = arctan ⁠y/x⁠ must be suitably defined, taking into account the correct quadrant of (x, y), as done in the equations above.

These formulae assume that the two systems have the same origin, that the spherical reference plane is the Cartesian xy plane, that θ is inclination from the z direction, and that the azimuth angles are measured from the Cartesian x axis (so that the y axis has φ = +90°).

Cylindrical coordinates (axial radius ρ, azimuth φ, elevation z) may be converted into spherical coordinates (central radius r, inclination θ, azimuth φ), by the formulas

These formulae assume that the two systems have the same origin and same reference plane, measure the azimuth angle φ in the same senses from the same axis, and that the spherical angle θ is inclination from the cylindrical z axis.

The following equations (Iyanaga 1977) assume that the colatitude θ is the inclination from the positive z axis, as in the physics convention discussed.

The del operator in this system leads to the following expressions for the gradient and Laplacian for scalar fields,

In the case of a constant φ or else θ = ⁠π/2⁠, this reduces to vector calculus in polar coordinates.

The physics convention . Spherical coordinates ( r , θ , φ ) as commonly used: ( ISO 80000-2:2019 ): radial distance r ( slant distance to origin), polar angle θ ( theta ) (angle with respect to positive polar axis), and azimuthal angle φ ( phi ) (angle of rotation from the initial meridian plane). This is the convention followed in this article.
The mathematics convention . Spherical coordinates ( r , θ , φ ) as typically used: radial distance r , azimuthal angle θ , and polar angle φ . + The meanings of θ and φ have been swapped —compared to the physics convention . The 'south'-direction x-axis is depicted but the 'north'-direction x-axis is not. (As in physics, ρ ( rho ) is often used instead of r to avoid confusion with the value r in cylindrical and 2D polar coordinates.)
In the mathematics convention : A globe showing a unit sphere , with tuple coordinates of point P (red): its radial distance r (red, not labeled); its azimuthal angle θ (not labeled); and its polar angle of inclination φ (not labeled). The radial distance upward along the zenith–axis from the point of origin to the surface of the sphere is assigned the value unity, or 1. + In this image, r appears to equal 4/6, or .67, (of unity); i.e., four of the six 'nested shells' to the surface. The azimuth angle θ appears to equal positive 90°, as rotated counterclockwise from the azimuth-reference x–axis; and the inclination φ appears to equal 30°, as rotated from the zenith–axis. (Note the 'full' rotation, or inclination, from the zenith–axis to the y–axis is 90°).
The output pattern of the industrial loudspeaker shown here uses spherical polar plots taken at six frequencies
Unit vectors in spherical coordinates