I have a set of points $(x_i, y_i)$ for which I want to determine the smoothness of the curve formed with them.
The horizontal distance $(x_{i+1} - x_i)$ between these points is not identical.
I have so far arrived at the formula below to calculate the smoothness of the curve, but am not sure if the result is likely to be correct and would appreciate feedback.
The concept is oriented at the calculation of total variation, but I am unsure whether I apply total variation properly. And the smaller the total variation, the smoother the set of points are.
If the points would be equidistant, total variation would be calculated as:
$$TV_{equidistant} = \sum_{i=0}^{n-1}\lvert{y_{i+1} - y_i}\rvert$$
(i. e.: take the absolute values of the vertical differences between the points and sum them up)
If the points are not horizontally equidistant, I assume that I should calculate the total variation in the following way:
$$TV_{{non-equidistant}} = \sum_{i=0}^{n-1}\lvert\frac{y_{i+1} - y_i}{(N-1)*\frac{x_{i+1} -x_i}{x_{0} -x_n}}\rvert$$
I. e.: each vertical difference between two points gets more relative weight, if these points are closer to each other on the x axis, as closer values on the x-axis mean less smoothness.
Furthermore these weights get multiplied with $(N-1)/(x_0 - x_n)$, so that with equidistant points I get the same result as with the formula before).
So my question is, whether this formula is a good measure for the smoothness of my non-equidistant points?