Spherical Coordinates

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spherical coordinates

[′sfir·ə·kəl kō′ȯrd·ən·əts]
A system of curvilinear coordinates in which the position of a point in space is designated by its distance 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 Spherical Coordinates 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 Spherical Coordinates 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

Figure 1

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

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