Spherical coordinate system

A point plotted using the spherical coordinate system

In Mathematics, the spherical coordinate system is a coordinate system for representing geometric figures in three dimensions using three coordinates, (ρ, φ, θ), where ρ represents the radial distance of a point from a fixed origin, φ represents the zenith angle from the positive z-axis and θ represents the azimuth angle from the positive x-axis. The geographic coordinate system is similar to the spherical coordinate system.

Coordinate system definition and notation

The spherical coordinate system represents points as a tuple of three components. For use in physical sciences and technology, the recommended international standard notation is r, ϑ, φ for distance, zenith and azimuth (ISO 31-11). (In the standard ϑ is preferred to θ, although it is the same Greek letter).

Otherwise, in America the components are typically notated as (ρ, φ, θ) for distance, zenith and azimuth, while elsewhere the notation is reversed for zenith and azimuth as (ρ, θ, φ). The former has the advantage of being most compatible with the notation for the two-dimensional polar coordinate system and the three-dimensional cylindrical coordinate system, while the latter has the broader acceptance geographically. The notation convention of the author of any work pertaining to spherical coordinates should always be checked before using the formulas and equations of that author. This article uses "American" notation.

The three coordinates (ρ, φ, θ) are defined as:

• 0 ≤ ρ is the distance from the origin to a given point P.
• 0 ≤ φ ≤ 180° is the angle between the positive z-axis and the line formed between the origin and P.
• 0 ≤ θ ≤ 360° is the angle between the positive x-axis and the line from the origin to the P projected onto the xy-plane.

φ is referred to as the zenith or colatitude, while θ is referred to as the azimuth.

According to this system, φ and θ lose significance when ρ = 0 and θ loses significance if sin(φ) = 0 (at φ = 0 and φ = 180°).

To plot a point from its spherical coordinates, go ρ units from the origin along the positive z-axis, rotate φ about the y-axis in the direction of the positive x-axis and rotate θ about the z-axis in the direction of the positive y-axis.

Coordinate system conversions

As the spherical coordinate system is only one of many three-dimensional coordinate systems, there exist equations for converting coordinates between the spherical coordinate system and others.

Cartesian coordinate system

Further information: Cartesian coordinate system

The three spherical coordinates are converted to Cartesian coordinates by:

<math>{x}=\rho \, \sin\phi \, \cos\theta \quad </math>

<math>{y}=\rho \, \sin\phi \, \sin\theta \quad </math>

<math>{z}=\rho \, \cos\phi \quad </math>

Conversely, Cartesian coordinates may be converted to spherical coordinates by:

<math>{\rho}=\sqrt{x^2 + y^2 + z^2}</math>

<math>{\phi}=\cos ^{ - 1} \left( {\frac{z}{{\sqrt {x^2 + y^2 + z^2 } }}} \right)</math>

<math>{\theta}=\tan ^{-1} \left( {\frac{y}{x}} \right)</math>

Geographic coordinate system

Further information: Geographic coordinate system

The geographic coordinate system is an alternate version of the spherical coordinate system, used primarily in geography though also in mathematics and physics applications. In geography, ρ is usually dropped or replaced with a value representing elevation or altitude.

Latitude is the complement of the zenith or colatitude, and can be converted by:

<math>{\delta}=90^\circ - \phi</math>, and

<math>{\phi}=90^\circ - \delta</math>,

though latitude is typically represented by φ as well. This represents a zenith angle originating from the xy-plane with a domain -90° ≤ φ ≤ 90°. The longitude is the azimuth angle shifted 180° from θ to give a domain of -180° ≤ θ ≤ 180°.

Cylindrical coordinate system

Further information: Cylindrical coordinate system

The cylindrical coordinate system is a three-dimensional extrusion of the polar coordinate system, with an h coordinate to describe a point's height above or below the xy-plane. The full coordinate tuple is (r, θ, h).

Spherical coordinates may be converted to cylindrical coordinates by:

<math> r = \rho \sin \phi \,</math>

<math> \theta = \theta \,</math>

<math> h = \rho \cos \phi \,</math>

Cylindrical coordinates may be converted to spherical coordinates by:

<math>{\rho}=\sqrt{r^2+h^2}</math>

<math>{\theta}=\theta \quad</math>

<math>{\phi}=\arctan\frac{r}{h}</math>

Applications

Spherical coordinates are useful in analyzing systems that are symmetrical about a point; a sphere that has the Cartesian equation x² + y² + z² = c² has the very simple equation ρ = c in spherical coordinates.

Spherical coordinates are the natural coordinates for physical situations where there is spherical symmetry. In such a situation, one can describe waves using spherical harmonics. Another application is ergonomic design, where <math>{\rho}</math> is the arm length of a stationary person and the angles describe the direction of the arm as it reaches out.

Just as the two-dimensional Cartesian coordinate system is useful on the plane, a two-dimensional spherical coordinate system is useful on the surface of the sphere. In this system, the sphere is taken as a unit sphere, so the radius is unity and can generally be ignored. The other two coordinates are as above. The latitude-longitude coordinate system is a variation of this system. This simplification can be very useful when dealing, for example, with objects such as rotational matrices (matrix rotation).

The concept of spherical coordinates can be extended to higher dimensional spaces and are then referred to as hyperspherical coordinates.

See also

• Cartesian coordinate system
• Cylindrical coordinate system
• Vector fields in cylindrical and spherical coordinates
• List of canonical coordinate transformations
• Hypersphere

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