Spherical coordinate system

In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are

the radial distance, r, along the line connecting the point to a fixed point called the origin;

the polar angle, θ, between this radial line and a given polar axis; and

the azimuthal angle, φ, which is the angle of rotation of the radial line around the polar axis.

(See graphic regarding the "physics convention".)

Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates.

The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle) is called the reference plane (sometimes fundamental plane).

Terminology

The radial distance from the fixed point of origin is also called the radius, or radial line, or radial coordinate. The polar angle may be called inclination angle, zenith angle, normal angle, or the colatitude. The user may choose to replace the inclination angle by its complement, the elevation angle (or altitude angle), measured upward between the reference plane and the radial line—i.e., from the reference plane upward (towards to the positive z-axis) to the radial line. The depression angle is the negative of the elevation angle. (See graphic re the "physics convention"—not "mathematics convention".)

Both the use of symbols and the naming order of tuple coordinates differ among the several sources and disciplines. This article will use the ISO convention frequently encountered in physics, where the naming tuple gives the order as: radial distance, polar angle, azimuthal angle, or

(

r

,

θ

,

φ

)

{\displaystyle (r,\theta ,\varphi )}

.