I apologize in advance for any imprecisions or badly defined concepts that I am not from the field.
I am working with experimental data, so the curves are defined by time series of $d$-dimensional coordinates (i.e. they are "trajectories" in $d$-dimensional spaces). I would like to measure and compare the "straightness" of these trajectories. As a first step, I have arclength reparametrized the trajectories such that consecutive points are at unit distance along the trajectory.
My current method is using the method from this post. That is:
- Get the tangent vector $\vec T_n =\frac{ x_{n+1} - x_n }{\lVert x_{n+1} - x_n \rVert} $ for all $n-1$ points $x$ where $x$ are $d$-dimensional coordinates.
- Sum $\lVert \vec T_{n+1} - \vec T_n \rVert $ for all $n-2$ tangent vectors $\vec T$.
- Divide the sum of 2. by total arclength.
If my interpretation is right, this should gives me a measure of how much the direction change along the curves, and thus how straight it is.
However, I am afraid that comparing the straightness of trajectories lying in space of different number of dimensions might introduce bias in this metric.
Is it possible (as in, well defined) to compare the straightness of space curves if they are embedded in space of different dimensions ?
EDIT: Added more details and changed "curvature" to "straightness".