# Forum > Discussion > Mad Science and Grumpy Technology >  Circles, waves and triangles.

## halfeye

I was trying to get to sleep.

So, if a circle rotates at the same speed as a sheet of paper (or whatever, so long as it can be drawn on), is passing it, and a marker marks the surface (the speed of rotation of the circle being such that when the marker is moving exactly in the direction of the paper, the speed of the marker is static with respect to the paper), what is the line that is drawn called?

It's not a sine wave, because in that the marker doesn't move forward or back with respect to the world the paper is passing through. It's not a series of semi-circles, because the angle of the line reaches 45 degrees at the middle height, which isn't true of a semi-corcle.

----------


## Radar

Not sure if this shape has a specific name or can be described with a simple function aside from a parametric curve like (here skipping a few adjustable parameters for simplicity)
x = cos(t) + t
y = -sin(t)

Update before I actually posted: I did some search and the shape is called a cycloid. There are some proper equations for the shape given in the link along with some other properties.

----------


## halfeye

> Not sure if this shape has a specific name or can be described with a simple function aside from a parametric curve like (here skipping a few adjustable parameters for simplicity)
> x = cos(t) + t
> y = -sin(t)
> 
> Update before I actually posted: I did some search and the shape is called a cycloid. There are some proper equations for the shape given in the link along with some other properties.


That does appear to be it, thanks very much, I looked on wikipedia but without knowing it's name I didn't find it.

----------


## Willie the Duck

> That does appear to be it, thanks very much, I looked on wikipedia but without knowing it's name I didn't find it.





> Update before I actually posted: I did some search and the shape is called a cycloid. There are some proper equations for the shape given in the link along with some other properties.


I am now curious what your research looked like, Radar. I, like halfeye, would have had lots of issues trying to tell a search engine what concept for which I was trying to find the term.

----------


## Radar

> I am now curious what your research looked like, Radar. I, like halfeye, would have had lots of issues trying to tell a search engine what concept for which I was trying to find the term.


My first connection was that the shape is a specific 2D projection of a helix, which was close enough for some discussions on the subject to mention a cycloid.

----------


## Rockphed

It is the shape a point on a wheel makes as the wheel rolls. When I put that in a search engine the Wikipedia page for "cycloid" shows up.

Now I am not sure I would have leapt to that without seeing "cycloid", so thanks to Radar.

One of the interesting things about cycloids is that ropes naturally form them when allowed to hang from their ends.

----------


## halfeye

> One of the interesting things about cycloids is that ropes naturally form them when allowed to hang from their ends.


That is a catenary, which is different from a parabola, and I am pretty sure it's not the same as a cycloid.

----------

