This week, I find myself having to remember how to graph in polar coordinates. I am talking about cardioids, rose petals, lemniscates and the like. It’s been a while…
There are many websites that will draw these for you. Desmos has polar mode. But I have not yet found a website that will show the graphs being constructed. But I remember reading a very nice explanation that (I think) was in one of Howard Anton’s calculus texts. If I am remembering this right, he said something like:
Imagine a clock with one hand that rotates around, connected to the x-y plane at the origin.
Then, imagine an ant (Anton’s Ant!) walking along that clock hand as it rotates. The ant’s distance from the origin is given by the function r=f(θ).
As the clock arm rotates, the ant moves in an out along the arm. The location of the ant traces out the graph of the polar function r=f(θ).
And note that if the radius is negative, that just means that the ant has walked “the other way” along the clock arm.
For example, these are three pictures showing the clock arm and the ant at successive times as we graph the cardioid r = f(θ) = 2 + 4 cos (θ):
I don’t know how to make my own applets, but I do know how to use Interactive Physics as a work-around. It’s not as good as a real applet. You can change parameters easily but if you want to change to a new function, that takes more work. And if you want to share with students and colleagues, they need the software, which is kind of pricey. (Though there is a way to get student versions quite reasonably. Email me if interested…totally legal!)
In any case, one thing you can do is export your experiments as videos. You lose the ability to vary parameters but at least you get the flavor. So that’s what I did…
A CARDIOID with an inner loop
A ROSE PETAL with 5 loops
If you do happen to have the Interactive Physics software package and would like the source files for these demos, post a comment or send me an email. That way, you can run these yourself on your own computer, changing the parameters to make discoveries.
Or just leave them to run for as long as you feel like. They are very soothing to watch.