# Polar Coordinates Several methods can be used to describe the location of a point on a plane, the most common being the Cartesian coordinate system. However, in some cases the polar coordinate system is much better suited to the task. **The polar coordinate system maps a point's distance from the origin of the coordinate system** (which corresponds to the magnitude of a complex number on the complex plane), **to the angle of the point's vector with the $x$ axis** (which corresponds to the argument of a complex number on the complex plane). Cartesian coordinates: $(x, y)$ Polar coordinates: $(r, \theta)$ The polar coordinates can be used to simplify descriptions of circular graphs. For example all of the below describe a [[circle]] with radius of 1: - $x^2+y^2 = 1$ - $y = \sqrt{1-x^2} \quad \cup \quad y = -\sqrt{1-x^2}$ - $r=1$ A circle with radius of 2: - $x^2+y^2 = 4$ - $y = \sqrt{4-x^2} \quad \cup \quad y = -\sqrt{4-x^2}$ - $r=2$ ## Converting Between Coordinates $x=r\cos\theta \quad y=r\sin\theta$ $ r=\sqrt{x^2+y^2} \quad \theta = \begin{cases} x\ne 0 & \arctan \left(\frac y x \right)\\ x=0 \land y\ge 0 & \frac \pi 2 \\ x=0 \land y < 0 & -\frac \pi 2 \end{cases} $