Polar coordinate

Polar coordinate system is a planar (2-dimensional) coordinate system in which each point on a plane is represented by its distance from a reference point (the pole or origin) and an angle from a reference direction (the polar axis).

TL;DR

• It uses the pair of numbers (distance,angle) to represent a point.
• We usually use the notation \((r,\theta)\).
• We usually require \(r \ge 0\) and \(\theta \in [0,2\pi)\) (there are other conventions out there).

How does it work?

In the demonstration below, move your mouse to see how points on the plane are represented in the polar coordinate system.

Just like the in the Cartesian system, each point on the plane is represented by a pair of numbers in the polar coordinate system. But instead of using \((x,y)\), the polar coordinate system uses \((r,\theta)\) where \(r\) represents the distance between this point and the pole while \(\theta\) angle between the reference direction and the ray formed by the pole and this point.

Polar coordinates are may not be unique

• First of all, \((0,\theta)\) for all choice of \(\theta\) point to the pole.

Conversion between Cartesian coordinate system

The polar coordinates \((r,\theta)\) can be converted to the Cartesian coordinates \((x,y)\) using the following equations:

\[ \begin{align*} x &= r \cos(\theta) \\ y &= r \sin(\theta) \end{align*} \]