How to define / characterize a curved line graph I have data that consist of multiple curved lines between point A and point B. I would like to categorize these lines into a number of groups.
The groups are:

*

*straight

*slight break

*heavy break

*early break

*late break

A visual interpretation of the lines can be found below:

However, I am having difficulty describing the lines / categories. I am only interested in the shape (not the length).
I tried rotating the images 90 degrees and describing them as PDFs of a beta distribution with different values for alpha and beta. However, this approach is not feasible for straight, slight break and hard break.
Any suggestions/ideas?
 A: You could try to evaluate the curvature of the lines, roughly speaking the second derivative of the points of the curves w.r.t. the arc-length.
Assuming that you know the points of the curves and you can introduce the arc-length variable, the parametric expression of a curve become $\mathbf{r}(s)$, and the curvature is $k(s) = |\mathbf{r}''(s)|$. Since you're interested only in the shape of the curves and not in their dimension, you can normalize them to get unit-length curves, so that $s \in [0, 1]$.
Once you have the function $k(s)$, you could evaluate both the "amount of curvature" of the trajectory and the "distribution in space" of the curvature along the trajectory:

*

*the amount of curvature gives you the distinction between slight and heavy, and could be computed as an example with the integral of $k(s)$ with $s \in [0,1]$, $K = \int_{0}^{1} k(s) ds$;

*the distribution of curvature could be evaluated as a sort of weighted mean of curvature over the length, as an example $M_1 = \frac{1}{K}\int_{0}^{1} s k(s) ds$ (something like a first momentum, but you could try high-order momentum as well), that is a valid expression for $K \ne 0$, i.e. for every line that is not straight (but for a straight line, with no curvature at all, it's meaningless to talk about curvature). With $M_1$ approaching $0$ (or at least $<1/2$) you could talk about Early Break, with $M_1$ approaching to $1$ (or at least $>1/2$) you could talk about Heavy Break.

Discretization. You can use the numerical method you like the most to get a discrete (or at least, finite dimensional) approximation of your continuous curves, and the numerical schemes that better fits the approximation method.
Comments. If you start with these kind of images, you may need to compensate for perspective, before performing any kind of computations about the trajectories.
