Definition:Spherical Coordinate System
Definition
A spherical coordinate system is a polar coordinate system in $3$ dimensions.
A distinct point $O$ is identified, and referred to as the origin.
Polar Axis
The polar axis of a spherical coordinate system is the vertical straight line which passes through the origin $O$.
Horizontal Axis
Having identified the polar axis of a spherical coordinate system, one then selects a distinct horizontal straight line, also passing through the origin $O$, perpendicular to the polar axis
This horizontal straight line is referred to as the horizontal axis.
Initial Meridian Plane
The initial meridian plane of a spherical coordinate system is the vertical plane in which the polar axis and horizontal axis both lie.
Hence, let $P$ be an arbitrary point $P$ in space.
Let $\mathbf r$ be the radius vector of $P$ with respect to $O$.
The position of $P$ is specified in spherical coordinates by:
- $(1): \quad$ the length of $\mathbf r$, that is, the distance of $P$ from the origin $O$, denoted by $r$.
- $(2): \quad$ the angle between $\mathbf r$ and the polar axis, known as the colatitude of $P$, denoted by $\theta$
- $(3): \quad$ the angle between the horizontal axis and the projection of $\mathbf r$ onto the horizontal plane, known as the longitude of $P$, denoted by $\phi$.
Also known as
A system of spherical coordinates is also known as a system of spherical polar coordinates.
Also see
- Results about spherical coordinates can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): spherical coordinate system
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): spherical coordinate system