If one considers a polygon as a discrete approximation to a continuous closed curve, then each vertex has influence over a whole section of the curve. So it is not unreasonable to be less local when assigning a direction and 'curvature' at the vertex, dependent on the neighboring vertices. Whether there's a natural way to do this (and perhaps determine a 'best" continuous curve being approximated) might be an interesting question. Not that I disagree with your notion of fairness. --ms On 18-Feb-17 20:23, Warren D Smith wrote:
Your rectangle example seems damning. ANY vector "curvature" which always has the direction of the angle-bisector, is going to be killed by your rectangle example.
And I really do not think it is fair to use any other direction! -- Unless you abandon all pretense this has anything to do with "motion of the boundary in the 'inward normal direction' (which is a locally-defined notion) with velocity proportional to 'curvature' (which is a locally-defined notion)."
So that's kind of cute, actually. I mean, the thing works for smooth curves, but depends upon smoothness, and it works for simplices, which are not smooth, but that is seen to be special.