Changing the Domain of Polar Graphs

In today’s post: a work-around for specifying the domain of a polar functions. (hint: it involves a quick switch to parametric equations.)

What’s your favorite type of equation? I have always loved polar equations, but in the last year or so, parametric equations have taken polar’s place as #1.  This post may help to explain one of the reasons behind my change of heart.

Graphing polar functions is super fun. The curves polar functions generate are fun and unexpected. And when you graph them on Desmos, they’re simply stunning. But as of right now, you cannot specify the domain of a polar function on Desmos.  As far as I can tell, it will only draw polar graphs for values of θ from 0 to 12π radians:

This suffices for many basic graphs, but even in the case of this spiral, it means that we aren’t seeing the whole picture.  For example, what happens when θ is negative? What about for larger values of θ?

Never fear! You CAN view a polar graph for any values of θ that you want, you just have to convert it to parametric first.

For my example of how to convert polar to parametric, I’ll use this pretty little curve.

If you’re unfamiliar with parametric equations, this is truly your lucky day.  These little guys are the bread and butter of so much of the amazing stuff that can be graphed on Desmos.  The idea is just that you have separate equations to define what x is doing and what y is doing.

In order to define x and y independently of each other, we will use the polar-to-rectangular conversion as follows:

Let’s do a quick substitution:

Desmos requires parametric equations to be written in terms of the parameter t and not θ, like you see below.  However, if you put them into Desmos like this, you won’t get the graph you expected at all.

In Desmos, parametric equations are written as an ordered pair, kind of like a point:

Notice that it allows you to specify a domain.  This is fantastic if you want to graph something where the curve won’t be complete after 12π radians.

Let’s take another look at that spiral, using parametric equations instead of polar.

Here it is for only positive values of θ, or t, after the polar-to-parametric conversion:

Now check it out with positive and negative values of t:

Pretty sweet!

Or let’s look at this polar function with a period of 17π. Here’s what Desmos will graph:

But if we convert this to parametric, we can see the entire curve!

One other benefit of switching polar to parametric is that in parametric form, one can do certain animations that wouldn’t have been possible in polar.

For example, check out this polar butterfly, made up of two mirror-image polar curves:

I wanted to make the wings flap, but it was going to be pretty hard in polar form, so I converted to parametric where I could single out the x-coordinates to make the wings beat.