[From Bruce Abbott (2016.08.22.2230 EDT)]
For those who may want to know how curvature formula that is used in power-law research was derived, there is a nice series of videos from the Khan Academy on that very topic. The first video provides a simple, non-mathematical treatment explaining the concept of curvature, and the second begins the mathematical treatment by illustrating how lines tangent to a curve at given points change angle as the points follow the curve. The rate at which the angle changes with the position of the point along the curve will be higher for tight curves and lower for shallower ones. Curvature is then defined as the change in tangent-vector (the angle of the unit tangent-line) dT per change in arc-length ds:
Ê = dT/ds
This treatment makes it clear that curvature reflects the rate of change of the tangent vector with the change in arc length; time per se is not involved, as Martin Taylor has demonstrated in several of his posts. This is logical, because curvature should not depend on how fast one follows along the curve, but rather, how much the curve bends per unit of change in position along it.
First video: Curvature Intuition: https://youtu.be/ugtUGhBSeE0
Second video: Curvature formula part 1: https://youtu.be/gspjhwSNMWs
Bruce