I wrote a short paper with Alex about the use of angular speed in asessing the speed-curvature power law. The power law comes in two forms: tangential speed and curvature (VC power law), and angular speed and curvature (AC power law).

I was wrong in my first paper on the power law to say that the two forms of the power law were equivalent. The exponents of the laws are equivalent - but the strength of the law is not, expressed as r or r^{2}. The correlation is much stronger in the AC power law, and for a trivial reason. The reason is that angular speed (A) depends on both curvature (C) and speed (V), we can calculate it as A=VC.

Correlating C to V is fine - it tells you if the object is systematically going slower in curves and faster in straight path segments or not. Speed is independent of curvature. This is what most researchers who study the speed-curvature power law do, so my paper is not very relevant for most of the published research on the power law. However, most papers state explicitly that the two forms of the power law are equivalent, and there is a minority of papers using the AC power law, some of them my own.

Here is the mistake: correlating C to A is like correlating C to VC. For most trajectories, curvature will be strongly correlated to angular speed - the speed of “turning” - simply because turning is faster in curves. We know this from A = VC.

For a more intuitive example, consider a set of rectangles of different sizes, and we want to know if the widths are heights of these rectangles are related - maybe they are all squares. We can correlate widths (w) are heights (h) of all the rectangles in the set to find this out. We will get a strong correlation only if there is a systematic relationship between w and h. On the other hand, if we try to correlate widths with the areas (a), where a=wh, we will get a very strong correlation in most cases, simply because small-width rectangles will tend to have small areas, and large- width rectangles will tend to have large areas. The widths are heights don’t have to be related at all.

In that example, speed and curvature are like width and height, and angular speed is like area.

Here is a link to the preprint: Angular speed should be avoided when assessing the speed-curvature power law of movement | bioRxiv