8. Curves in Polar Coordinates

r = sin (2θ) − 1.7

This is a real graph using polar coordinates.
Okay, I admit to adding the eye and smile. :-)

Later, on this page...

Don't miss the Polar graphs interactive applet.

We'll plot the graphs in this section using a computer. You'll also learn how to sketch some of them on paper because it helps you understand how graphs in polar coordinates work.

Don't worry about all the difficult-looking algebra in the second part of the answers - it's just there to demonstrate that polar coordinates are much simpler than rectangular coordinates for these graphs. We convert them using what we learned in the last section, Polar Coordinates.

Helpful Background

Curves in polar coordinates work very similarly to vectors. See:

Vector concepts


Need Graph Paper?

rectangular grid
Download graph paper

(Polar graph paper included.)

Sketch each of the following functions using polar coordinates, and then convert each to an equation in rectangular coordinates.

Example 1: r = 2 + 3 sin θ

(This polar graph is called a limacon from the Latin word for "snail".)

Here's another example of a limacon:

Example 2: r = 3 cos 2θ

Example 3: r = sin θ − 1

(This one is called a cardioid because it is heart-shaped. It is a special case of the limacon.)

Continues below

Example 4: `r = 2.5`

Example 5: r = 0.2 θ

This is an interesting curve, called an Archimedean Spiral. As θ increases, so does r.

See also Equiangular Spiral.

Later, we'll learn how to find the Length of an Archimedean Spiral.

Example 6: r = sin (2θ) − 1.7

This is the face I drew at the top of this page. We're not even going to try to find the equivalent in rectangular coordinates!

You can play with this graph in the following interactive applet.

Interactive Graph

You an explore the above graphs using this interactive graph.

Use the slider below the graph to trace out the curves.

See what happens as you go beyond the normal domain for these graphs (i.e. when `theta` is less than `0` or greater than `theta = 2pi`).

Change the function using the select box at the top of the graph.

Choose function:


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Check out Polar Coordinates and Cardioid Microphones for an application of polar coordinates.