curve parameterized by arc length

There are some differences between a curve and a function. Consider the functions \(f(t) = (t, 2t)\) and \(g(t) = (2t, 4t)\). Both of them trace out the same curve, but \(f\) does it faster. We would like to take derivatives without having to worry about how the curve was parameterized. We can do this by taking the curve 'parameterized by arc length' or considering a function that moves along the curve at unit speed. That is, "one unit of time = one unit of arc length".