I'm looking for a way to define some sort of a smoothness value for each point on a line. Then I would in turn define the smoothness of a closed loop to be the smoothness of its least smooth point.
Maybe "smoothness" is not the right term here. The intuition I have in my mind is a loop that is made from steel wire or some similar flexible material. Tension inside the wire would cause the curvature of the loop to be roughly the same at each point. If I cut the loop open, the steel would return to be just a straight line segment.
To demonstrate what I mean, take these equations for example:
$$ \begin{align} x & = \sin{2t} \cos{t} \\ y & = \sin{2t} \sin{t} \end{align} $$
On the plot below, it seems like the curves around the centre point are smoother, while the curves around the four edges are less smooth.
For instance, if I drew a tangent at the edges, it would seem like I'm getting away from the edges faster as I move along the tangent, compared to if I drew the tangent through the $x$ or $y$ axis.
At the same time, if I draw a circle, the smoothness value appears to be the same at each point.
So, my questions would be:
- Is there a name for this "smoothness value" that I am thinking of?
- If so, how do I calculate it at each point for a given set of equations, e.g. for my four-leaf clover above?
- How would I continuously deform my line towards a local maximum of such a smoothness value? Basically, how do I smooth out those edges?
I'd give the second two questions a stab myself but I'm stuck with number one.