Yesterday Zorn Leung conjectured that if a plane curve has curvature , then it is contained in a circle (of some radius). I will give an example to show that this is actually false.
The example is the following curve:
This curve is called the prolate cycloid (I’ve just learned this name today). Geometrically it is the path traced out by a fixed point at a radius (here ) , where is the radius of a rolling circle (the animation is not of the same scale as my example because I am too lazy to make the animation myself, I just download it from the net):
It is easy to calculate that
It is not necessary to calculate exactly the minimum value of , just observe that is periodic and that for all , this implies for some constant .
But it is easy to see that is unbounded, and in particular it cannot be contained in any circle, no matter how large we choose the radius.
Remark 1 The conjecture should be true if the curve is simple closed and the radius of the circle can be chosen to be . I haven’t written down the full details of the proof, though. I think it would be a nice exercise for the geometry students ☺