Arc length parametrisation

Definition

Let $\gamma \colon [a, b] \to \mathbb{R}^n$ be a regular $C^1$ curve. Its arc length function is $$s(t) = \int_a^t \|\gamma'(u)\| \, du, \qquad t \in [a, b],$$ and its length is $\ell = s(b)$. Since $s'(t) = \|\gamma'(t)\| > 0$, the function $s \colon [a, b] \to [0, \ell]$ is a strictly increasing $C^1$ bijection with a $C^1$ inverse $t(s)$. The arc length parametrisation of $\gamma$ is the curve $\tilde\gamma \colon [0, \ell] \to \mathbb{R}^n$, $\tilde\gamma(s) = \gamma(t(s))$.

Remarks

The curve $\tilde\gamma$ has the same image and direction as $\gamma$, and $\|\tilde\gamma'(s)\| = 1$ for each $s$. So $\tilde\gamma'(s)$ is the unit tangent vector. The length of $\tilde\gamma$ from $\tilde\gamma(s_0)$ to $\tilde\gamma(s_1)$ is $s_1 - s_0$. For the helix $\gamma(t) = (\cos t, \sin t, t)$, $\|\gamma'(t)\| = \sqrt{2}$, so $s = \sqrt{2}\, t$ and $\tilde\gamma(s) = (\cos(s / \sqrt{2}), \sin(s / \sqrt{2}), s / \sqrt{2})$. The arc length parametrisation is unique up to a shift of the parameter. It simplifies the formulas for curvature.

Explanations

L1 human

Reparametrise the curve so that the parameter is the distance travelled from the start. Then the point moves at constant speed $1$, and only the shape of the curve remains.