We are accustomed to investigate such points (of high curvature) in various graphs presenting experimental data, see for example the spectral peaks' maxima. But this question concerns not the extremal points but all the other ones with high curvature. Do you know examples where they are useful, i.e. carry some important (physical or otherwise) information? What are they? Comments concerning inflection points (i.e. those with minimum curvature) are also welcome. Note, however, that finding maximum curvature points is only possible when units in both directions are identical (or "x" and "y" are dimensionless, of course).
In the field of computer vision corner points (high curvature) always carry maximum information and are used for find similarities between objects. For example two different views of an object might be differently scaled, skewed, rotated and some portions may be occluded so a direct match does not work. Depending on the perceived distortion, one needs to match more and more number of points to detect the distortion. A single pint match (maximal point) will work only if the object has only 1 prominent feature. If the transformation is only perspective at least 4 high curvature points are needed to correctly identify the transformation. Given a confidence level, which translates to threshold of curvature, more the number of points, better is the match.
Computer vision is tightly related to physical observation and the a set of high texture points (not just maximal) contain highest information about any observation. On the other hand if the set becomes very large to contain low curvature points, it becomes erroneous to match. The set of high curvature points is called as features of the data.
In terms of physics features of the observation means number of dimensions of the observation. If the underlying phenomena is low dimension one will see lower number of high curvature points. If the underlying phenomena is high dimension, the number of high curvature points will increase and the model has to include higher number of dimension.
Hope this helps you.
Thank you, Shalender Singh, for your comment. My problem with it is that there are high curvature points, but I don't see clearly the curves they belong to.
Dear Shalender,
as you are familiar with problems in relativity, I will give you an example from this area. As we know, relativistic mass may be written as m/m0 = 1 / \sqrt {1-(v/c)2}. This curve, with x=v/c and y=m/m0, has -a bit surprisingly - highest curvature at v/c \approx 0.51, although visually we would rather locate it at v/c near 0.9. At v/c=0.51 the moving mass increases roughly by 15% in respect to its rest value. No 'nice' numbers, but perhaps we can roughly say that relativistic effects surely cannot be neglected when v>0.51c? And the classical mechanics should work satisfactorily below this value?
Marek -
Perhaps I am missing the point, but are you asking, in your answer/example/clarification above, if it may be the case that because the greatest instantaneous rate of increase in mass occurs when v = 0.51 c, might this be a good breakpoint for whether or not to be able to use classical mechanics? That is an interesting question. I would suppose that would depend upon the specific application and the accuracy needed.
I see that your question is with regard to the physical interpretation of high curvature points in general, as opposed to the usual emphasis on extremal points that in physics, statistical science, operations research (such as the old simplex method), and other areas of application have perhaps an often clearer interpretation. My guess is that the practicality of such a physical interpretation of highest curvature may vary wildly by topic/application. In economics I suppose it could be interpreted as the point of 'diminishing returns.' ???Perhaps your applications could be considered in a manner analogous to that concept, if I am not misunderstanding your topic.
I'm not certain regarding the comment about the same units or dimensionless x and y being needed. If you rewrote your relativity example with x=v and y=m, wouldn't you come to the same conclusions regarding the highest curvature?
Cheers - Jim
James,
the dimensionless units are rather necessary, if you think in Euclidean, geometrical terms. This is because for flat curves, given as y=f(x), you have:
(signed) curvature = y'' / [1+ (y')2]3/2,
where y' = dy/dx, and y'' = d2y/dx2. You can approximate curvature quite well by y'' at extrema (where y'=0), but this approximation doesn't work correctly neither for inflection points nor for "knees". Especially those "knees" are what I'm asking for.
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