To describe a differentiable curve, one could use a parameterisation. Imaginatively, one could imagine driving a car with the objective of aliging the car’s path with the curve. One has unlimited control over the car’s throttle (the velocity) and “ability to turn” (the centripetal acceleration, one could imagine a car that can turn on-the-spot)

The ability to control the car’s throttle gives rise to a lot of variability in how you can drive the car. Suppose you were asked to drive a car in a straight line. You could go slowly, quickly, or alternate the throttle.

However if the throttle was fixed, then all you can do is to choose to either go forwards or backwards. At no moment can you choose to turn as it would cause you to swerve away from the straight line. Similarly, a differentiable curve may have a lot of parametrisations, but it would only have “two” arc-length parametrizations.

Curvature

The curvature of a curve is analagous to the magnitude of the centripetal acceleration of a projectile that’s moving with constant velocity. To see this, firstly we know that we can parametrize all types of uniform circular motion using \(\mathbf{r}(t) = (R \cos (kt), R \sin (kt))\) where \(R\) is the radius and \(k\) is the angular velocity. Note that since the velocity has magnitude kept constant, the acceleration is always perpendicular to the velocity (Check that if \(\| \mathbf{r}'\|\) is constant then \(\mathbf{r}' \cdot \mathbf{r}'' = 0\))

If we want the velocity vector to be a unit vector, we require \(\| \mathbf{r}'(t) \| = Rk = 1\). As such, this forces the projectile to take the path \(\mathbf{r}(t) = (R \cos (\frac{1}{R}t), R \sin (\frac{1}{R}t))\) instead.

Now the acceleration vector would have magnitude \(\|\mathbf{r}''(t)\| = \frac{1}{R}\), which coincides with the curvature of a circle with radius \(R\).

Torsion

In \(\mathbb{R}^2\), the curvature alone almost uniquely determines an arc-length parametrisation curve with some caveats. For example, the starting point and velocity of the curve being undetermined, whether the curve goes clockwise or anticlockwise and whenever the curvature is zero at some point.

The nice thing about “turning” in \(\mathbb{R}^2\) is that you can only turn clockwise or anticlockwise with some magnitude, as such the curvature alone is mostly sufficient. However, you can turn in all sorts of directions in \(\mathbb{R}^3\). The change in the direction you’re turning is called the torsion. Once you account for that, we arrive at the fundamental theorem of curves which roughly states that “every regular curve in three-dimensional space, with non-zero curvature, has its shape (and size or scale) completely determined by its curvature and torsion.”