Parameterizing a curve by arc length

To parameterize a curve by arc length, the procedure is

1. Find the arc length $s(t)$ . If the curve is regular then $s(t)$ is a monotonically increasing function.
2. Calculate the inverse of the arc length $t(s)$ . This is another monotonically increasing function.
3. Set the new parameter $u = t(s)$ . Now $\alpha(u)$ has arc length parameterization.

Intuitively, if the curve if ``moving'' quickly in some region, then its arc length will increase rapidly and the inverse $t(s)$ will increase proportionally slowly. In the parameterized curve $\alpha(u)$ these two factors will cancel out and the curve will increase in length uniformly. What is the velocity of $\alpha(u)$ ? We have

$$\alpha'(u) = \ensuremath{\frac{d}{d s}}\alpha(t(s))$$ (2.1)

$$= \alpha'(t) t'(s)$$ (2.2)

$$= \alpha'(t) / s'(t)$$ (2.3)

$$= \alpha'(t) / \vert \alpha'(t) \vert,$$ (2.4)

where we have used the fact that $\ensuremath{\frac{d}{d s}}t(s) = 1 / \ensuremath{\frac{d}{d t}} s(t)$ when $s$ and $t$ are inverses of each other, and the definition of arc length in equation (2.4). Hence we have found that the velocity $\vert\alpha'(u)\vert$ is one for an arc length parameterized curve.