# What is a polar plot?

Jan 22, 2018

See below.

#### Explanation:

Consider a typical plot that you will have came across before:

$y = {x}^{2}$

The plot looks like this:

graph{x^2 [-2.47, 2.53, -0.36, 2.14]}

In this plot, every value along the $x$ axis is linked to a point on the $y$ axis. At $x = 2$, $y = 4$. We write the coordinates as: $\left(2 , 4\right)$. When we write coordinates in the form $\left(x , y\right)$ we call them Cartesian coordinates.

In polar coordinates we write the coordinates of a point in the form $\left(r , \theta\right)$ where $r$ is the distance directly between the point and the origin and $\theta$ is the angle made between the positive $x$-axis and that line.

The diagram below provides a simple illustration of how a point can be expressed in either Cartesian or polar coordinates.

From this we can also see how to convert between polar and Cartesian coordinates using simple trigonometry:

By Pythagoras theorem:

${r}^{2} = {x}^{2} + {y}^{2}$

and also:

$\theta = {\tan}^{- 1} \left(\frac{y}{x}\right)$

To convert back into Cartesian:

$x = r \cos \theta$
$y = r \sin \theta$

So a polar plot is quite simply plot where the function has been written in polar form, (i.e. a function that links $r$ to $\theta$ as appose to a function that links $y$ to $x$).

Whether or not you wish to use polar coordinates really depends on the situation. If the graph has some form of circular symmetry then perhaps polar may be advantageous over Cartesian.

Some examples of polar plots:

This is an ellipse with equation:

r(theta) = 0.75/(1-0.5cos(theta)

At an angle of ${60}^{o}$ from the x-axis this would have a value:

$r \left({60}^{o}\right) = \frac{0.75}{1 - 0.5 \cos \left({60}^{o}\right)} = 1$

So a line drawn from the origin at 60 degrees from the $x$-axis will meet the ellipse when the length of that line is 1.

Polar plots can also be used to produce some interesting spirals as well,

This graph has equation: $r \left(\theta\right) = {e}^{\sqrt{\theta}}$

As you can imagine this would be considerably difficult to work with in Cartesian.

But anyway, that is general idea of a polar plot.