Concepts in Curvature

Step 3: Understand dT/ds

Our next step is to find the rate of change of the unit tangent vector with respect to the position of a point moving along the curve $dT/ds$. We need to find this with respect to the curve instead of with respect to time because the curvature of the curve is not dependant on how fast a particle is traveling along the curve. We're finding geometry, not trajectory.

To find $dT/ds$, we need to find the rate of change of the unit tangent vector with respect to time, $dT/dt$, and then divide it by the rate of change of the position of the particle with respect to time, $ds/dt$. The $dt$s cancel out, leaving us with $dT/ds$.

We already know how to find $ds/dt$ from the previous step, so all we need to do is find $dT/dt$. The math for this gets even messier than before, so just make sure you understand what we're doing conceptually.

The graph to the right shows both our unit tangent vector $T$ in white, as well as $dT/ds$ in blue. You can also see how $dT/ds$ is "pulling" $T$ as time increases by toggling the switch below.