A pen is attached to the wheel. Assume the pen is a sphere

Factual Questions
1

OK. I don’t know how I’m going to explain this.

Observing a bicycle with a reflector on the wheel drive by the other night, I was looking at the reflector go round and round. And I wondered: If that reflector had a pen that drew a line on a piece of paper beside the bicycle, what would the resultant “wave” look like? Would it be some sort of modified sine wave? Would the line cross over itself at the bottom when the pen goes from moving in the direction of the bike to away from it? How much would the distance from the center of the wheel affect things?

2

Depends on the size of the wheel, right?

3

I believe it should just look like a simple scalloped edge line.

4

If the pen in on the rim it would draw a sine wave. If at the hub a straight line. In between a series of loops (sort of like a cursive “l” or “el) whose amplitude diminishes as the pen is moved from the rim to the hub.

5

Surely there’s a Spirograph component that would illustrate this.

What do you mean by “the pen is a sphere?”

6

On the rim, it seems to draw a cycloid. Inside the rim, it’s a curtate trochoid, assuming this is what you mean:

Spirograph was the first thing I though of, but the various wheels stay within the page, in a circle, rather than moving along the road.

7

Oh, that’s the word…. I knew there was a better word than “scallop”…

8

Wow. Some quick responses. I’m not surprised it has a name.

Just a little joke

ETA: Though the Wiki version isn’t (IMO) as good as the one I herard. In the version I heard the punchline is "The physicist presents his results. ‘First, we assume the cow is a sphere…’” and lets the listener/reader infer the rest.

9

an animation:

10

Something like this?

11

Kinda related… if you make the “pen” out of an LED light strip and take a long exposure, you get something like this:

From a persistence of vision bike light project: Overview | Bike Wheel POV Display | Adafruit Learning System (edit: fixed link)

12

Here’s a spirograph simulator - Spirograph pattern generator - ScienceDemos.org.uk

You can try out some other options.

13

Imagine how much fun Euclid would have had with a spirograph.

14

My recollection of Spirograph kits was that in addition to the hollow gears with teeth on the inside perimeter, the kits also included a couple of straight-line gear racks totaling maybe 12 or 18” of length.

So you could tack that straight track to your paper and use one of the convex circular gears with the teeth on the outside perimeter to roll along it. Which would be a strong analog to the bicycle wheel experiment. The smaller the gear you chose, the more revolutions you could get out of your e.g. 18” run.

15

I don’t think I ever had, or at least never used, the straight-line spirograph thing.

16

I expect that over time they got more elaborate. And most likely started out with a basic kit and a “deluxe” kit with more gizmos.

So depending on which year you got yours, etc., you may never have seen the gear racks.

17

Probably in the early 70s? Anyway, that animator is really cool.

18

Closer to ABS(SIN)

19

Yup,

20

By the way, they recently relaunched the Spirograph toys at Target: Spirograph products at Target (along with a whole host of Gen-X/Millennial nostalgia toys). There’s an endcap display that takes you right back to the 80s: