Chapter 3

Around an Elliptical World

Be warned fellow traveller. In this chapter, we venture outside the world of the spirograph into that mysterious and frequently terrifying world of theory. First, we will explore a way of apporximating the path traced by a spirograph with an elliptical track. But in the next chapter, we will not only make this approximation exact, we can generalize our equations to make a spirograph with any track imaginable.

For our first method, instead of starting from scratch, let’s break down the equations we got for a circular hypotrochoid from Chapter 1. We can then edit them to describe an elliptical track.

Let’s rewrite this out so we have three terms:

The first term of both equations describe the track: a circle with a radius of . The second term gives us an “offset from the track.” In other words, the first 2 terms describe where the center of the rotating circle will be. The third term describes the rotation of the point around the center of the rotating circle.

Now, to start replacing these terms to describe our elliptical track. But to describe this track, we must first ask:

What Is an Ellipse?

For our purposes, an ellipse is just a squashed circle. Being a squashed circle, it no longer has 1 radius, but needs to be described with the biggest radius and the smallest radius, called the semimajor and semiminor axes respectively.

graph of an ellipse showing the semimajor and semiminor axis

Let the semimajor axis have a length of and the semiminor have a length of . (these are different from points and from Chapter 1, we ran out of sensible letters for our variables). The parametric of an ellipse is

Wait, why a ? Notice how we’re making the distinction between and ? Before, we were using as our input to our parametric equations. But now that we’re also defining our track as a parametric equation, we need to decouple “time” from “angle”. This doesn’t change anything with the equations, but we changed it here to cause less confusion.

Replacing this parametric in for our first term gets us:

A Saving Approximation

Second problem: we’ve replaced the s in the first term, but what about for the last term? Before it was the radius of the track, but now there is no consistent radius for the track. What is this even used for again? Well, remember when we said that the arc from A to C must be equal to the arc from C to B?

diagram demonstrating that rolling requires the length travelled around the track equals the amount the tracing wheel rolls

That idea still holds true since we’re still rolling the tracing wheel! Calculating the exact length of that arc for an ellipse gets more complicated, so for now we can introduce an approximation. We might not know the exact arc length of the ellipse, but we can know the perimeter of the ellipse:

Finding the radius of a circle with this perimeter will give us a good enough approximation for .

Perimeter of an ellipse side tangent What we show above is known as the elliptic integral, and it gives off the accurate value for the perimeter. It’s derived from finding the length of the parametric curve, which we discuss a little in the next chapter. The elliptic integral is famous because it can’t be written in elementary functions, but that hasn’t stopped mathematicians from trying!

Finding Our Center

We’ve done so much math but we haven’t had a single Desmos for this chapter. Let’s fix that and graph what we have!

Hey! That looks pretty good! Wait, what is that?!

What’s what? Try making the track super squished by making large and small. Making larger too might make it easier to see. Try it with different rotations along the track by changing .

Answer

The rotating circle is clipping through the track!

The problem is that we still haven’t edited that second term, meaning the center of our rotating wheel isn’t correct. Let’s look at a diagram from Chapter 1 when we were working with a circular track:

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

Notice how the point of tangency (point ) is collinear with the centers of the two wheels (points and ). This means that the parametric function perfectly describes the position of that center point.

Compare this with a diagram of an elliptical track:

diagram with an elliptical track and the starting and ending positions of the tracing wheel

The point of tangency and the two centers are no longer collinear, so the equation no longer describes the center of the rotating circle. We’ll need something more sophisticated. To find the offset of from , let’s find the slope of .

diagram with an elliptical track with trangles showing the position M relative to point C

To find this slope, (equal to ), notice how that it’s perpendicular to the line tangent with the ellipse at point (equal to ). And finding the slope of the tangent line is nothing a little calculus (and our favorite trig identities) can’t handle!

All we need to find now is the lengths of segments and . I hope you’re not trigged out:

Awesome! All that’s left to do is to replace our offset terms with these and we’re good to roll!

(don’t those equations just look lovely)

Playing Around

What’s that, is it Desmos-time?

Notice how the wheel kinda slips when it meets the pointer parts of the ellipse. This is because of that approximation we did earlier, but is something we can fix in the next chapter!