In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are

the radial distance, , along the line connecting the point to a fixed point called the origin; the polar angle, , between this radial line and a given polar axis; and the azimuthal angle, , which is the angle of rotation of the radial line around the polar axis.

(See graphic regarding the "physics convention".)

Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle) is called the reference plane (sometimes fundamental plane).

Terminology

The radial distance from the fixed point of origin is also called the radius, or radial line, or radial coordinate. The polar angle may be called inclination angle, zenith angle, normal angle, or the colatitude. The user may choose to replace the inclination angle by its complement, the elevation angle (or altitude angle), measured upward between the reference plane and the radial linei.e., from the reference plane upward (towards to the positive z-axis) to the radial line. The depression angle is the negative of the elevation angle. (See graphic re the "physics convention"not "mathematics convention".)

Both the use of symbols and the naming order of tuple coordinates differ among the several sources and disciplines. This article will use the ISO convention frequently encountered in physics, where the triple of coordinates are listed in the naming order: (radial distance, polar angle, azimuthal angle), denoted by . (See graphic re the "physics convention".)

Confusingly, the conventions in many mathematics texts use a different naming order: (radial distance, azimuthal angle, polar angle), denoted by or switching both the list order of the angle coordinates and the meaning of the letters and . Other conventions may also be used, such as for the radius from the z-axis, while using for radius from the origin. The mathematics convention and has the advantage that its letters are consistent with cylindrical coordinates in space and polar coordinates in the plane.

According to the conventions of geographical coordinate systems, positions are measured by latitude, longitude, and height (altitude). There are a number of celestial coordinate systems based on different fundamental planes and with different terms for the various coordinates. The spherical coordinate systems used in mathematics normally use radians rather than degrees; (note 90 degrees equals radians). And these systems of the mathematics convention may measure the azimuthal angle counterclockwise (i.e., from the south direction -axis, or 180°, towards the east direction -axis, or +90°)rather than measure clockwise (i.e., from the north direction x-axis, or 0°, towards the east direction y-axis, or +90°), as done in the horizontal coordinate system.Duffett-Smith, P and Zwart, J, p. 34. (See graphic re "mathematics convention".)

The spherical coordinate system can be seen as an extension of the planar polar coordinate system to three-dimensional space, and it can be similarly extended to define a hyperspherical coordinate system in -dimensional space.

Definition To define a spherical coordinate system, one must designate an origin point in space, , and two orthogonal directions: the zenith reference direction and the azimuth reference direction. These choices determine a reference plane that is typically defined as containing the point of origin and the x and yaxes, either of which may be designated as the azimuth reference direction. The reference plane is perpendicular (orthogonal) to the zenith direction, and typically is designated "horizontal" to the zenith direction's "vertical". The spherical coordinates of a point then are defined as follows:

The radius or radial distance is the Euclidean distance from the origin to . The inclination (or polar angle) is the signed angle from the zenith reference direction to the line segment . (Elevation may be used as the polar angle instead of inclination; see below.) The azimuth (or azimuthal angle) is the signed angle measured from the azimuth reference direction to the orthogonal projection of the radial line segment on the reference plane. The sign of the azimuth is determined by designating the rotation that is the positive sense of turning about the zenith. This choice is arbitrary, and is part of the coordinate system definition. (If the inclination is either zero or 180 degrees (= radians), the azimuth is arbitrary. If the radius is zero, both azimuth and inclination are arbitrary.)

The elevation is the signed angle from the x-y reference plane to the radial line segment , where positive angles are designated as upward, towards the zenith reference. Elevation is 90 degrees (= radians) minus inclination. Thus, if the inclination is 60 degrees (= radians), then the elevation is 30 degrees (= radians).

In linear algebra, the vector from the origin to the point is often called the position vector of P.

Conventions Several different conventions exist for representing spherical coordinates and prescribing the naming order of their symbols. The 3-tuple number set denotes radial distance, the polar angle"inclination", or as the alternative, "elevation"and the azimuthal angle. It is the common practice within the physics convention, as specified by ISO standard 80000-2:2019, and earlier in ISO 31-11 (1992).

As stated above, this article describes the ISO "physics convention"unless otherwise noted.

However, some authors (including mathematicians) use the symbol ρ (rho) for radius, or radial distance, φ for inclination (or elevation) and θ for azimuthwhile others keep the use of r for the radius; all which "provides a logical extension of the usual polar coordinates notation". As to order, some authors list the azimuth before the inclination (or the elevation) angle. Some combinations of these choices result in a left-handed coordinate system. The standard "physics convention" 3-tuple set conflicts with the usual notation for two-dimensional polar coordinates and three-dimensional cylindrical coordinates, where is often used for the azimuth.

Angles are typically measured in degrees (°) or in radians (rad), where 360° = 2 rad. The use of degrees is most common in geography, astronomy, and engineering, where radians are commonly used in mathematics and theoretical physics. The unit for radial distance is usually determined by the context, as occurs in applications of the 'unit sphere', see applications.

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 planeas 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.

+ Major conventions coordinates set order corresponding local geographical directions right/left-handed right right left Note: Easting (), Northing (), Upwardness (). In the case of the local azimuth angle would be measured counterclockwise from to .

Unique coordinates Any spherical coordinate triplet (or tuple) specifies a single point of three-dimensional space. On the reverse view, any single point has infinitely many equivalent spherical coordinates. That is, the user can add or subtract any number of full turns to the angular measures without changing the angles themselves, and therefore without changing the point. It is convenient in many contexts to use negative radial distances, the convention being , which is equivalent to or for any , , and . Moreover, is equivalent to .

When necessary to define a unique set of spherical coordinates for each point, the user must restrict the range, aka interval, of each coordinate. A common choice is:

radial distance: polar angle: , or , azimuth : , or .

The azimuth is sometimes restricted to the half-open interval , or radians, which is the standard convention for geographic longitude, instead of the interval .

For the polar angle , the range (interval) for inclination is , which is equivalent to elevation range (interval) . In geography, the latitude is the elevation.

Even with these restrictions, if the polar angle (inclination) is 0° or 180°elevation is −90° or +90°then the azimuth angle is arbitrary; and if is zero, both azimuth and polar angles are arbitrary. To define the coordinates as unique, the user can assert the convention that (in these cases) the arbitrary coordinates are set to zero.

Plotting To plot any dot from its spherical coordinates , where is inclination, the user would: move units from the origin in the zenith reference direction (z-axis); then rotate by the amount of the azimuth angle () 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 angle.

Applications

Just as the two-dimensional Cartesian coordinate system is usefulhas a wide set of applicationson a planar surface, a two-dimensional spherical coordinate system is useful on the surface of a sphere. For example, one sphere that is described in Cartesian coordinates with the equation can be described in spherical coordinates by the simple equation . (In this systemshown here in the mathematics conventionthe sphere is adapted as a unit sphere, where the radius is set to unity and then can generally be ignored, see graphic.)

This (unit sphere) simplification is also useful when dealing with objects such as rotational matrices. 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.

Three dimensional modeling of loudspeaker output patterns can be used to predict their performance. A number of polar plots are required, taken at a wide selection of frequencies, as the pattern changes greatly with frequency. 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 equationsthe Laplace and the Helmholtz equationsthat arise in many physical problems. The angular portions of the solutions to such equations take the form of spherical harmonics. Another application is ergonomic design, where is the arm length of a stationary person and the angles describe the direction of the arm as it reaches out. The spherical coordinate system is also commonly used in 3D game development to rotate the camera around the player's position

In geography

Instead of inclination, the geographic coordinate system uses elevation angle (or latitude), in the range (aka domain) and rotated north from the equator plane. Latitude (i.e., the angle of latitude) may be either geocentric latitude, measured (rotated) from the Earth's centerand designated variously by or geodetic latitude, measured (rotated) from the observer's local vertical, and typically designated . The polar angle (inclination), which is 90° minus the latitude and ranges from 0 to 180°, is called colatitude in geography.