A curve made up of lines

I've seen some yarn art made by stretching the yarn over nails set up like this:

I think people have assumed when you do something like the above that makes a circle, but you can see it goes outside of a perfect circle, though it is close.

Mathematically, the dark blue curve outside the circle results when you connect the points:
[a, c] and [c, c-a] for some c as a varies from 0 to c (pictured is c=5).

Dr. Rose found the parametric equation for the curve:

I tried to find the y=f(x) but it turned out it was 5 pages long!
The graph of the radius in polar coordinates for this function is:

Which looks a lot like a normal distribution but is too short and wide for what would be the standard deviation. You can see the radius is always greater than 5, so it does indeed lie outside the circle of radius 5.