# Spherical Coordinates

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## spherical coordinates

[′sfir·ə·kəl kō′ȯrd·ən·əts]*r*from the origin or pole, called the radius vector, the angle φ between the radius vector and a vertically directed polar axis, called the cone angle or colatitude, and the angle θ between the plane of φ and a fixed meridian plane through the polar axis, called the polar angle or longitude. Also known as spherical polar coordinates.

## Spherical Coordinates

The spherical coordinates of a point *M* are the three numbers *r*, θ, and ɸ. These numbers are defined relative to three mutually perpendicular axes, which are called the x-axis, *y-axis*, and z-axis and intersect at the point *O* (Figure 1). The number *r* is the distance from *O* to *M*. The number θ is the angle between the vector and the positive direction of the r-axis. The number ɸ is the angle through which the positive half of the x-axis must be turned counterclockwise so that it coincides with the vector where *N* is the projection of *M* on the ry-plane. The spherical coordinates of *M* thus depend on the choice of *O* and the three axes. The relations between the spherical and Cartesian coordinates are given by the following equations:

*x = r* sin θ cos ɸ

*y = r* sin θ cos ɸ

*z = r* cos θ

Spherical coordinates are widely used in mathematics and in applications of mathematics to physics and engineering.