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.
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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)).