Introduction of Polar Coordinates

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

When we think about plotting points in the plane, we usually think of rectangular coordinates $(x,y)$ in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. Polar coordinates are points labeled $(r,θ)$ and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.

The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is called the angular coordinate, polar angle, or azimuth. The radial coordinate is often denoted by $r$ or $ρ$ , and the angular coordinate by $ϕ$, $θ$, or $t$.

Examples of Polar Coordinates

Points in the polar coordinate system with pole $0$ and polar axis $L$. In green, the point with radial coordinate $3$ and angular coordinate $60$ degrees or $(3,60^{\circ})$. In blue, the point $(4,210^{\circ})$.

Polar Graph Paper

A polar grid with several angles labeled in degrees

Angles in polar notation are generally expressed in either degrees or radians ($2\pi $ rad being equal to $360^{\circ}$). Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics and mathematical physics.  In many contexts, a positive angular coordinate means that the angle $ϕ$ is measured counterclockwise from the axis.  In mathematical literature, the polar axis is often drawn horizontal and pointing to the right.

Plotting Points Using Polar Coordinates

The polar grid is scaled as the unit circle with the positive $x$-axis now viewed as the polar axis and the origin as the pole. The first coordinate $r$ is the radius or length of the directed line segment from the pole. The angle $θ$, measured in radians, indicates the direction of $r$. We move counterclockwise from the polar axis by an angle of $θ$,and measure a directed line segment the length of $r$ in the direction of $θ$. Even though we measure $θ$ first and then $r$, the polar point is written with the $r$ -coordinate first. For example, to plot the point $(2,\frac{\pi }{4})$,we would move $\frac{\pi }{4}$ units in the counterclockwise direction and then a length of $2$ from the pole. This point is plotted on the grid in Figure.

Plotting a point on a Polar Grid

Plot of the point $(2,\frac{\pi }{4})$,by moving $\frac{\pi }{4}$ units in the counterclockwise direction and then a length of $2$ from the pole.

Uniqueness of polar coordinates

Adding any number of full turns ($360^{\circ} $ or $2\pi$ radians) to the angular coordinate does not change the corresponding direction. Also, a negative radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction. Therefore, the same point can be expressed with an infinite number of different polar coordinates($r, \phi \pm n\cdot 360°$) or ($−r, \phi \pm (2n + 1)\cdot 180°$), where $n$ is any integer. Moreover, the pole itself can be expressed as ($0, ϕ$) for any angle $ϕ$.