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Arc Length, Curvature, and Related

The motivation for this study is to generalize the notion of trip along a path, where we wish to remove the notion of where we are traveling and focus more on how we are traveling. If we had a very high sample rate, like 88,200 Hz, for Cartesian coordinates \(x\) and \(y\) we could do a simple linear vibrational spectra analysis which could make a good distinguishing signature for the vehicle being driven, but we have a very low sample rate, 1 Hz, and low resolution (~ 0.1 meters) and noisy (at least due to rounding) to boot.

Arc Length

If we ride in a wheeled vehicle the arc length that we travel is proportional to the number of rotations that the wheels turn, assuming that the wheels to not slip on the road. Arc length can be thought of as the total distance traveled when we straighten out the roads, or the total distance traveled "not as the crow flies."

We defining arc length in 2-D (\(x\) and \(y\)); it would appear that it is not, in general, simple analytic function of \(x\) and \(y\). It's not possible to write \[ s = s(x,y) \mathrm{,} \label{s_xy_wrong}\] arc length, \(s\) as a function of displacements in Cartesian coordinates \(x\) and \(y\) of any other coordinate system. In general \(s\) can have more than one value at a given \(x\),\(y\) value. There may be very specific cases when we can write a form like [\(\ref{s_xy_wrong}\)], but never in general. Arc length, \(s\), tends to be defined in terms of either a differential equation, integral equation, or as a parameter in a parametric vector equation. All of them work the same. We have the Pythagorean differential form \[ \mathrm{d} s ^2 = \mathrm{d} x^2 + \mathrm{d} y^2 \, \mathrm{,} \] the integral form \[ s = \int_{t^\prime=0}^t \left[ \left(\frac{\mathrm{d}x}{\mathrm{d}t^\prime}\right)^2 + \left(\frac{\mathrm{d}y}{\mathrm{d}t^\prime}\right)^2 \right] \mathrm{d} t^\prime \, \mathrm{,} \label{s_int_xy} \] where we integrate over time \(t^\prime\); and the parametric vector equation, if we can write it, is \[ \vec{r}(s) = x(s)\,\hat{x} + y(s)\,\hat{y} \, \mathrm{,} \] where \(\hat{x}\) is the constant unit vector in the \(x\) direction, and \(\hat{y}\) is the constant unit vector in the \(y\) direction. Playing around with the these equations there's lots of other forms that that we can have.

It's clear that arc length is a purely a geometrical quantity and it can be defined by just spacial constructs, but in our calculations of it we are required to use the speed of object in it's travel. Why?

We'll use [\(\ref{s_int_xy}\)] given \(x(t)\) and \(y(t)\) as data at fixed time \(t\) steps, to get \(s(t)\) at the same fixed time, \(t\), steps.

Curvature

We have chosen arc length as one degree of freedom. We choose curvature, \(\kappa\), as the generated coordinate compliment to arc length, \(s\). We in 2-D motion we need transformation from Cartesian coordinates \(x, y\) to \(s, \kappa\). At any point in the path the change in \(s\) and \(\kappa\) are perpendicular.

We can express \(\kappa\) is the magnitude a vector \[ \kappa = \frac{\left|\dot{\vec{r}} \times \ddot{\vec{r}}\right|} {\left| \dot{\vec{r}}\right|^3} \] where \(\hat{n}\) is the unit normal vector, \(\dot{\vec{r}}\) is the velocity vector, and \(\ddot{\vec{r}}\) is the acceleration vector. Note that \(\kappa\) is positive if we travel in a counter-clock-wise circle and negative if we travel in a clock-wise circle. In 2-D, we can write \(\kappa\) in terms of \(x\) and \(y\) as \[ \kappa = \frac{\dot{x}\,\ddot{y} - \dot{y}\,\ddot{x}} { \sqrt{\left( \dot{x}^2 + \dot{y}^2 \right)^3}} \mathrm{.} \]

Dividing by \(\left|\dot{\vec{r}}\right|^3\) is not so appealing. We see that in order to go in a path that is not straight we must accelerate, so why not use a measure like the component of acceleration that is perpendicular to the velocity like where the acceleration is the vector sum of the of two components \[ \ddot{\vec{r}} \equiv \ddot{r}_T\,\hat{T} + \ddot{r}_n\,\hat{n} \] where \(\hat{T}\) is the unit tangent vector that is in the direction of \(\dot{\vec{r}}\) and \(\hat{n}\) is the unit vector in the direction that we are curving toward. We have \[ \ddot{r}_n = \frac{ \left|\dot{\vec{r}} \times \ddot{\vec{r}}\right|} {\left|\dot{\vec{r}}\right|} \] giving \[ \ddot{r}_n = \frac{\dot{x}\,\ddot{y} - \dot{y}\,\ddot{x}} {\sqrt{\dot{x}^2 + \dot{y}^2}}\mathrm{.} \] It can be shown that \[ \ddot{\vec{r}} = \frac{\mathrm{d}^2 s}{\mathrm{d} t^2} \,\hat{T} + \left(\frac{\mathrm{d} s}{\mathrm{d} t}\right)^2 \,\kappa\, \hat{n} \, \mathrm{.} \]

References