Arc Length
For a given parametric equation as a position function, arc length is speed. It can be evaluated for a given time as an arc length.
Description
For a position function expressed as a parametric equation like r(t) = f(t)i + g(t)j + h(t)k, it is well understood that the first derivative is velocity and the second derivative is acceleration.
The speed of movement along the function is given by:
For a given time, the speed function evaluates to the distance traveled by the original function in that time: arc length.
Note that notation is not standardized. Sometimes it uses a curly v, to indicate its relation to the velocity vector (i.e., v(t) = r'(t)). a is already reserved for the acceleration vector (i.e., a(t) = v'(t) = r''(t)).
Arc Length Parameterization
In some cases, it is useful to re-parameterize a parametric equation in terms of the arc length. For example, it can be useful to calculate the tangent vector for a given distance along an arc rather than for a given time.
This is accomplished by rewriting the above function of t. Substitute t with du, and integrate arc length from time 0 to time t with respect to u.
This can be simplified down to a statement like s(t) = α du, which can of course be rewritten as s = α t, but more importantly can be rewritten as t = s/α. This last formulation can be substituted back into r(t) to give r(s), the arc length parameterization.