Chapter 2

Drawing Outside the Lines

Now in many spirograph kits, the large track gears also have teeth on the outside. This means (given enough precision and patience), that we can move the rotating circle on the outside of the track! This will form an epitrochoid.

We could do what we did in the last chapter, setting up a new diagram and doing a lot of trigonometry. However, there’s a simpler way of doing this.

Let’s take a look at our final equation for a hypotrochoid:

Remember that:

Notice that for an epitrochoid, the objective is still to find the tracing point , and that , , and will remain unchanged. So how can we tell an equation that we want the rotating wheel to be on the outside of the fixed gear?

diagram showing the circular track, starting position of the tracing wheel, and the ending position of the tracing wheel

Here’s that old Desmos from last time. See if you can find out how to get the rotating gear on the outside of the fixed gear.

Hint 1 Remember that the radius of the rotating wheel is the only variable that needs to change.

Hint 2 Making the radius of the rotating wheel bigger isn’t going to work. What else can we do with the radius?

Answer Try making negative and play around with ! See what happens!

Ok! So making the radius negative is the same as moving the rotating wheel on the outside of the fixed wheel. But why does this work?

To gain some intuition for why this might be true, let’s graph both the hypotrochoid and epitrochoid of the same track on the same Desmos.

You might notice some things:

  1. The wheels rotate in opposite directions.
    • This makes sense since they’re making contact on 2 different sides of the track.
  2. They stick together through thick and thin.
    • This is a consequence of how we’re defining our parametric equation. Both wheels are given the same (the rotation of the rotating wheel around the track).
    • For us, this means that for any given , the amount of track covered for both wheels is the same.
  3. They both rotate the same amount.
    • Remember in Chapter 1 how we said that the rotation of the tracing circle is based on the amount of track covered? Based off of our last observation, this means that both wheels rotate the same amount.

Taking a look at our equation shows that making our radius negative changes both the position and direction of rotation of the inner wheel:

From a Negative Radius

So you can make the radius negative to make an epitrochoid. But what if you wanted a positive to correspond to an epitrochoid? Well, that’s nothing a little algebra can’t solve.

Let’s first start with our hypotrochoid equation but replace with . (And don’t forget those trig identities):

Playing Around

As always, a A Desmos a day keeps the doctors away: