# Definition:Polar Coordinates

## Contents

## Definition

**Polar coordinates** are a technique for unique identification of points on the plane.

A distinct point $O$ is identified.

### Pole

The point $O$ is referred to as the **pole** of the polar coordinate plane.

### Polar Axis

A ray is drawn from $O$, usually to the right, and referred to as the **polar axis**.

## Identification of Point in Plane with Ordered Pair

Let $P$ be any point different from $O$.

Let a straight line $OP$ be drawn from $O$ to $P$.

### Radial Coordinate

The length of $OP$ is called the **radial coordinate** of $P$, and usually labelled $r$.

### Angular Coordinate

The angle measured anticlockwise from the polar axis to $OP$ is called the **angular coordinate** of $P$, and usually labelled $\theta$.

If the angle is measured clockwise from the polar axis to $OP$, its value is considered negative.

The ordered pair $\tuple {r, \theta}$ is referred to as the **polar coordinates** of $P$.

In order to distinguish them from those in Cartesian coordinates, points in **polar coordinates** are often found denoted within angle brackets: $\polar {r, \theta}$.

## Polar Coordinate Plane

A plane upon which a system of polar coordinates has been applied is known as a polar coordinate plane.

## Historical Note

Polar coordinates were invented by Jacob Bernoulli in $1691$.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 5$: Trigonometric Functions - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Polar Form of Complex Numbers - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $6$: Curves and Coordinates