Spherical coordinate system

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In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its inclination angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. The inclination angle is often replaced by the elevation angle measured from the reference plane.

The radial distance is also called the radius or radial coordinate, and the inclination may be called colatitude, zenith angle, normal angle, or polar angle.

In geography and astronomy, the elevation and azimuth (or quantities very close to them) are called the latitude and longitude, respectively; and the radial distance is usually replaced by an altitude (measured from a central point or from a sea level surface).

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

Contents

Definition

To define a spherical coordinate system,one must choose two orthogonal directions, the zenith and the azimuth reference, and an origin point in space. These choices determine a reference plane that contains the origin and is perpendicular to the zenith. The spherical coordinates of a point P are then defined as follows:

  • the radius or radial distance is the Euclidean distance from the origin O to P.
  • the inclination (or polar angle) is the angle between the zenith direction and the line segment OP.
  • the azimuth (or azimuthal angle) is the signed angle measured from the azimuth reference direction to the orthogonal projection of the line segment OP on the reference plane.

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