Definition:Polar Coordinates

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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.


PolarCoordinates.png


The ordered pair $\left({r, \theta}\right)$ 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: $\left\langle{r, \theta}\right\rangle$.


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