Regular Curves and Arc Length

Arc-length parametrization is the entry point to the Frenet-Serret Frame and the definition of curvature for space curves — curvature is literally defined as $|\beta''(s)|$ for a unit-speed curve $\beta$, a formula that is meaningless without first fixing the unit-speed normalization. In computer graphics and CAD, arc-length reparametrization (often computed numerically since closed forms rarely exist, as in Example 2) is essential for uniform-speed animation along a spline path and for font and curve rendering where equal parameter increments must correspond to equal visual distances. In robotics and motion planning, a robot arm's end-effector trajectory is frequently specified as a regular curve, and re-timing it to constant speed (or to speeds respecting velocity/acceleration limits) is a direct engineering use of these ideas. In physics, the arc-length parameter is the natural "proper time"-like variable for a particle's spatial trajectory, decoupling geometric shape from the dynamics that produced a particular time-parametrization. In cartography and geodesy — Gauss's own domain — computing the length of a geodesic path on a curved surface reduces, after a choice of local coordinates, to exactly this rectification integral.

:::faq[What happens at endpoints of a closed interval, like $\alpha : [a,b] \to \mathbb{R}^n$?] The construction goes through with minor bookkeeping: one-sided derivatives are used at $a$ and $b$, and regularity means $\alpha'(t) \neq 0$ for all $t \in [a,b]$, including these one-sided derivatives at the endpoints. The arc-length function $s : [a,b] \to [0, L]$ (where $L$ is the total length) is then a strictly increasing bijection between closed intervals, and the Inverse Function Theorem applies on the interior with continuity extending to the closed endpoints. This is exactly the setting of the corollary on total arc length, which is the version most often used in practice: computing the length of a curve segment between two specified endpoints.

Ask the tutor: How does the arc-length reparametrization construction change when the curve lives on a surface instead of in $\mathbb{R}^n$, and what replaces the ordinary derivative $\alpha'(t)$ there?

Think of $\alpha(t)$ as the position of a particle at time $t$. The trace is the visible track left behind — a curve drawn in space — but the parametrization also records how the particle moves along that track: fast, slow, forward, backward, pausing, or even reversing direction. Two very different parametrizations can have the identical trace. For instance $\alpha(t) = (t, t^2)$ and $\beta(t) = (t^3, t^6)$ both trace the parabola $y = x^2$, but $\beta'(0) = (0,0)$ — the particle described by $\beta$ stops at the origin, even though geometrically the parabola has a perfectly good tangent line there. Regularity is exactly the condition that forbids this kind of hidden stop: it says the particle's velocity is never the zero vector, so the direction of motion is always defined and the trace has a genuine, well-defined tangent at every parameter value. Once we know $\alpha'(t) \ne 0$ everywhere, the speed function $v(t) = |\alpha'(t)|$ is a smooth, strictly positive function of $t$. Integrating speed over time gives distance traveled — this is exactly single-variable calculus, "distance = $\int$ speed $\,dt$" — and because speed is strictly positive, the resulting arc-length function $s(t)$ is strictly increasing. A strictly increasing smooth function with nonvanishing derivative has a smooth inverse (this is the Inverse Function Theorem in one variable), so we may solve for $t$ as a smooth function of $s$, and substitute back into $\alpha$ to get a new parametrization $\beta(s) = \alpha(t(s))$, traversing the exact same trace, but now at unit speed: $|\beta'(s)| = 1$ for all $s$. This unit-speed parametrization is the geometer's canonical clock, in which "time elapsed" and "distance traveled" coincide.

Example 1 (a genuinely non-circular curve: logarithmic spiral). Let $\alpha : \mathbb{R} \to \mathbb{R}^2$ be given by

a logarithmic spiral, with basepoint $t_0 = 0$ (so $\alpha(0) = (1,0)$). Regularity. Compute

Then

using $(\cos t - \sin t)^2 + (\sin t+\cos t)^2 = 2\cos^2t + 2\sin^2 t = 2$. So

Since this is never zero, $\alpha$ is regular on all of $\mathbb{R}$. Arc length from $t_0 = 0$.

Check monotonicity directly: $s'(t) = \sqrt{2}e^t > 0$, confirming Step 1 of the theorem without needing to re-derive it. As $t$ ranges over $\mathbb{R}$, $s(t) = \sqrt2(e^t-1)$ ranges over $J = (-\sqrt2, \infty)$, an open interval, as predicted. Inverting. Solve $s = \sqrt2(e^t - 1)$ for $t$:

This is smooth on $J$ since the argument of the logarithm, $1 + s/\sqrt2$, is strictly positive there. Unit-speed reparametrization.

As a check on the theorem's conclusion, differentiate using the chain rule (writing $r(s) = 1 + s/\sqrt2$, so $r'(s) = 1/\sqrt2$, and $\theta(s) = \ln r(s)$, so $\theta'(s) = r'(s)/r(s) = \frac{1}{\sqrt2\, r(s)}$):

So $\beta'(s) = \frac{1}{\sqrt2}(\cos\theta - \sin\theta,\ \sin\theta+\cos\theta)$, and

exactly as guaranteed: $|\beta'(s)| = 1$ for every $s \in (-\sqrt2,\infty)$.

Example 2 (a curve where regularity must be checked, not assumed: a cubic with an asymmetric coefficient). Let $\alpha : \mathbb{R} \to \mathbb{R}^3$,

(This is chosen asymmetrically — the coordinates have different degrees and a non-unit coefficient — precisely so no cancellation is a coincidence of symmetry.) Regularity. $\alpha'(t) = (2t,\ 2t^2,\ 1)$. The third coordinate is identically $1 \ne 0$, so $\alpha'(t) \ne (0,0,0)$ for every $t \in \mathbb{R}$: the curve is regular everywhere, with no need for case analysis. Speed.

Notice $4t^4 + 4t^2 + 1 = (2t^2+1)^2$ (check: $(2t^2+1)^2 = 4t^4 + 4t^2 + 1$ ✓.) So

since $2t^2 + 1 > 0$ always. This is a clean example where the speed simplifies to a perfect square under the root — worth verifying explicitly rather than assuming. Arc length from $t_0 = 0$.

Check: $s'(t) = 2t^2+1 > 0$ for all $t$, consistent with regularity; $s$ is strictly increasing on all of $\mathbb{R}$, and since $s(t) \to \pm\infty$ as $t \to \pm \infty$, $J = s(\mathbb{R}) = \mathbb{R}$. Here $s(t) = \frac{2}{3}t^3 + t$ is a strictly increasing cubic, and while it is a bijection $\mathbb{R}\to\mathbb{R}$ (odd degree, positive derivative everywhere), it has no elementary closed-form inverse — solving $\frac23 t^3 + t = s$ for $t$ requires Cardano's cubic formula. This is worth stating plainly: the theorem guarantees $t(s)$ exists and is smooth, purely from $s'(t) = 2t^2+1 \neq 0$, without requiring us to write it down explicitly. This is the practical value of the existence theorem — arc-length parametrization is guaranteed to exist for every regular curve even when it cannot be written in closed form. We can still verify the unit-speed property implicitly: writing $\beta(s) = \alpha(t(s))$,

so $|\beta'(s)| = 1$ identically, exactly as in Example 1 — but here confirmed by the general argument rather than by explicit formulas, illustrating why the theorem's proof (not just its examples) is the essential content.