# 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**