Submitted by Djohariyea:
I saw this gif floating around on my dashboard
It’s a straight bar skewed at what looks like a 45 degree angle? As it rotates through the vertical plane, it traces that curve. I was wondering if that curve has is a piece of a certain conic section? It kinda looks like a hyperbola to me, but I was wondering if you had any insights based on how the mechanism works, if that’s a clue to what that curve is…
Hi, Djohariyea! Firstly, I’d like to apologize for waiting so long to respond to your submission. When I still had school, I simply did not have time to deal with my inbox, but now it’s summer! I know your question could be answered with one sentence, but I’m going to a full-blown explanation for followers who may want it.
Alright, this is stuff you need to know to understand what’s going on:
- Hyperbola (it’s a mathematical curve, one of the conic sections)
- Hyperboloid of one sheet (it’s a three-dimensional shape obtained by revolving a hyperbola around its semi-minor axis. It is a ruled surface.)
- Ruled Surface (a surface where for every point on that surface, there is a straight line that lies on that surface).
What’s going on?
Look at the straight bar in the gif. More importantly, look at the 3D shape the bar seems to be tracing out in the air as it spins. Hopefully, you’ll notice that it’s making this sort of shape:
It’s a hyperboloid. A hyperboloid is a ruled surface, meaning that it’s curved shape can be created using straight lines. Have you ever heard of a mathematical envelope, or string art? Same idea. Here is a picture of a hyperboloid with it’s ruled surface much more apparent:
In this hyperboloid I’ve just showed you, the curved, hour-glass-like shape is made completely out of straight lines (string!). (This particular hyperboloid is actually double-ruled because there are lines going both up and down, criss-crossing.) But what do hyperboloids and ruled surfaces have to do with the gif? Look at the gif right now and imagine that the bar leaves a string behind, suspended in the air, at every single position that it is in during one rotation. It’s going to look a lot like the string hyperboloid above. What I am trying to say is that the tilted bar’s motion, as it revolves, takes the place of all the straight lines that make up a hyperbola. In other words, one moving straight line can act as many. This is what I mean, and this is what is going on in the submitted gif:
So, to finally address you original question, what the heck is that curve that the straight bar passes through? I think it is this:
And what is the name of the curve? Well, you can obtain a hyperboloid by rotating a hyperbola around its semi-minor axis, so I believe that the curve is a hyperbola. Here’s a gif illustrating this idea:
The End. Thanks for the submission!
Sources of all pictures in order of appearance, disregarding the submission: Hyperboloid 1, 2, 3, 4, and 5.
TL;DR: The bar in the gif makes a hyperboloid and the curve the bar passes through is a hyperbola.