 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: the radial distance of a point from a fixed origin, the zenith angle from the positive z-axis, and the azimuth angle from the positive x-axis. Image File history File links This is a lossless scalable vector image. ...
Image File history File links This is a lossless scalable vector image. ...
Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ...
In broad terms, the zenith is the direction pointing directly above a particular location (perpendicular, orthogonal). ...
Azimuth is the horizontal component of a direction (compass direction), measured around the horizon, from the north toward the east (i. ...
Notation Several different conventions exist for representing the three coordinates. In mathematics, the components are typically notated as (ρ, θ, φ) for radial distance, azimuth, and zenith, respectively. In physics, the notation for zenith and azimuth are reversed as θ is used to denote the zenith angle and φ is used to denote the azimuthal angle, in accordance with the International Standards Organization (ISO 31-11). The former has the advantage of being most compatible in the meaning of θ with the notation for the two-dimensional polar coordinate system and the three-dimensional cylindrical coordinate system, while the latter has broader acceptance geographically. Other notation uses r for radial distance.[1] 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 the "mathematical" convention. ISO 31-11 is the part of international standard ISO 31 that defines mathematical signs and symbols for use in physical sciences and technology. ...
A polar grid with several angles labeled in degrees In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. ...
2 points plotted with cylindrical coordinates The cylindrical coordinate system is a three-dimensional coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually denoted ) which measures the height of a point above the plane. ...
Definition The three coordinates (ρ, θ, φ) are defined as: - ρ ≥ 0 is the distance from the origin to a given point P.
- 0 ≤ θ < 2π is the angle between the positive x-axis and the line from the origin to the P projected onto the xy-plane.
- 0 ≤ φ < π is the angle between the positive z-axis and the line formed between the origin and P.
θ is referred to as the azimuth, while φ is referred to as the zenith, colatitude or polar angle.
θ and φ and lose significance when ρ = 0 and θ loses significance when 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 obtained from Cartesian coordinates by: Fig. ...
Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...
   Note that the arctangent must be defined suitably so as to take account of the correct quadrant of y / x. The atan2 or equivalent function accomplishes this for computational purposes. Atan2 is a two-parameter function for computing the arctangent in the C programming language. ...
Conversely, Cartesian coordinates may be retrieved from spherical coordinates by:
  
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. Map of Earth showing lines of latitude (horizontally) and longitude (vertically), Eckert VI projection; large version (pdf, 1. ...
A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ...
Latitude  is the complement of the zenith or colatitude, and can be converted by:
 , or , 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 measured in degrees east or west from 0°, so its domain is -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). Image File history File links This is a lossless scalable vector image. ...
Image File history File links This is a lossless scalable vector image. ...
2 points plotted with cylindrical coordinates The cylindrical coordinate system is a three-dimensional coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually denoted ) which measures the height of a point above the plane. ...
A polar grid with several angles labeled in degrees In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. ...
Cylindrical coordinates may be converted into spherical coordinates by:
   Spherical coordinates may be converted into cylindrical coordinates by:   
Applications The geographic coordinate system applies the two angles of the spherical coordinate system to express locations on Earth, calling them latitude and longitude. 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 a sphere. In this system, the sphere is taken as a unit sphere, so the radius is unity and can generally be ignored. This simplification can also be very useful when dealing with objects such as rotational matrices. Map of Earth showing lines of latitude (horizontally) and longitude (vertically), Eckert VI projection; large version (pdf, 1. ...
This article is about the geographical term. ...
Longitude is the east-west geographic coordinate measurement most commonly utilized in cartography and global navigation. ...
Fig. ...
A rotation matrix is a matrix which when multiplied by a vector has the effect of changing the direction of the vector but not its magnitude. ...
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. An example is in solving a triple integral with a sphere as its domain. To meet Wikipedias quality standards, this article or section may require cleanup. ...
The surface element is
 The volume element is  Spherical coordinates are the natural coordinates for describing and analyzing physical situations where there is spherical symmetry, such as the potential energy field surrounding a sphere (or point) with mass or charge. Two important partial differential equations, Laplace's equation and the Helmholtz equation, allow a separation of variables in spherical coordinates. The angular portions of the solutions to such equations take the form of spherical harmonics. In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ...
In mathematics, Laplaces equation is a partial differential equation named after its discoverer, Pierre-Simon Laplace. ...
The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation where is the Laplacian, is a constant, and the unknown function is defined on three-dimensional Euclidean space R3. ...
In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to re-write an equation so that each of two variables occurs on a different side of the equation. ...
In mathematics, the spherical harmonics are the angular portion of an orthogonal set of solutions to Laplaces equation represented in a system of spherical coordinates. ...
Another application is ergonomic design, where ρ is the arm length of a stationary person and the angles describe the direction of the arm as it reaches out.
The concept of spherical coordinates can be extended to higher dimensional spaces and are then referred to as hyperspherical coordinates. 2-sphere wireframe as an orthogonal projection Just as a stereographic projection can project a spheres surface to a plane, it can also project a 3-spheres surface into 3-space. ...
Notes Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ...
Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ...
is the 299th day of the year (300th in leap years) in the Gregorian calendar. ...
MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...
Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ...
is the 100th day of the year (101st in leap years) in the Gregorian calendar. ...
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