Polar Graphs

Now let's turn our attention graphs of polar functions and equations. Let's start by looking at some common types of polar graphs and then how polar graphs compare to the same function graphed as a rectangular function. Lastly, we will look at several specific types of polar graphs.

The following graph includes some specific examples of several common types of polar graphs. Select each equation to see what the graph looks like. Try to summarize what you see.

Summary

Here are a few things to consider for the polar graphs above. To help make some general statements, let \(c\) be a real number.

The polar equation \(r = c\) represents a circle centered at the origin having a radius of \(c\). The equivalenet rectangular equation is \(x^2 + y^2 = c^2\).
The polar equation \(\theta = c\) represents a line passing through the origin having a slope of \(\tan{c}\). The equivalent rectangular equation is \(y = \tan(c) x\).
The polar equation \(r = c \csc{\theta}\) represents a horizontal line equivalent to \(y = c\).
The polar equation \(r = c \sec{\theta}\) represents a vertical line equivalent to \(x = c\).
The polar equation \(r = c \sin{\theta}\) represents a circle centered at \(\left(0, \frac{c}{2}\right)\) having a radius of \(\frac{c}{2}\). This means that the circle will extend upwards along the \(+y\) axis if \(c > 0\) or it will extend downward along the \(-y\) axis if \(c < 0\).
The polar equation \(r = c \cos{\theta}\) represents a circle centered at \(\left(\frac{c}{2}, 0\right)\) having a radius of \(\frac{c}{2}\). This means that the circle will extend outwards along the \(+x\) axis if \(c > 0\) or it will extend outwards along the \(-x\) axis if \(c < 0\).
The polar equation \(r = c\,\theta\) is a spiral expanding outward from the origin in a counterclockwise direction. The spiral begins in the first quadrant if \(c > 0\) or in the third quadrant if \(c < 0\). The spacing between consecutive curls of the spiral is always \(2\pi \cdot c\).

Special Graphs

Limaçons are graphs defined by the polar equations \(r = a \pm b \sin{\theta}\) or \(r = a \pm b \cos{\theta}\) for Real numbers \(a\) and \(b\).

Roses are graphs defined by the polar equations \(r = a \sin(n\theta)\) or \(r = a \cos(n\theta)\) for Real numbers \(a\) and \(n\).

Lemniscates are graphs defined by the polar equations \(r^2 = a^2 \sin{2\theta}\) or \(r^2 = a^2 \cos{2\theta}\) for Real number \(a\).

Polar Coordinates

Polar Graphs

Special Graphs