Arc Length and Reparametrization

A parametrized curve comes with a built-in speedometer: the speed \lVert \mathbf{r}'(t) \rVert. If you know your speed at every instant, then the total distance you travel is just the accumulated speed — speed integrated over time. That single idea is the whole of arc length.

L = \int_a^b \lVert \mathbf{r}'(t) \rVert \, dt.

This is the same formula you met for the length of a graph — see arc length by integration — now written in the cleaner vector language: chop the curve into tiny pieces, each of length \lVert \mathbf{r}'(t)\rVert\,dt (speed × little time), and add them all up.

The arc-length function

Instead of only the total length, track how much length has been laid down by time t. Fix a starting time a and define the arc-length function

s(t) = \int_a^t \lVert \mathbf{r}'(u) \rVert \, du.

It measures the distance travelled along the curve from the start up to parameter t. By the Fundamental Theorem of Calculus, differentiating an integral with respect to its upper limit gives back the integrand:

\frac{ds}{dt} = \lVert \mathbf{r}'(t) \rVert = v(t).

So ds/dt is the speed — a clean restatement of "distance accumulates at a rate equal to speed." For a regular curve the speed is always positive, so s(t) is strictly increasing: it never doubles back, and therefore it can be inverted. That invertibility is the door to the most natural way of all to describe a curve.

Unit-speed (arc-length) parametrization

Because s(t) is strictly increasing, we can turn the story around and use the distance travelled, s, as the parameter instead of time t. Invert to get t(s) and feed it back in:

\boldsymbol{\gamma}(s) = \mathbf{r}\big(t(s)\big).

This reparametrized curve has a magical property. By the chain rule and ds/dt = \lVert \mathbf{r}'\rVert,

\boldsymbol{\gamma}'(s) = \mathbf{r}'(t)\,\frac{dt}{ds} = \frac{\mathbf{r}'(t)}{\lVert \mathbf{r}'(t)\rVert} = \mathbf{T}, \qquad \lVert \boldsymbol{\gamma}'(s)\rVert = 1.

A curve parametrized by arc length moves at unit speed: it covers one unit of length for every unit of parameter. Its velocity is the unit tangent \mathbf{T}. Nothing is wasted on bookkeeping how fast the point moves — all the remaining information is pure shape.

Every regular curve can be reparametrized by arc length. In the arc-length parameter s:

Time is arbitrary — you can traverse the same track slowly, quickly, or in fits and starts, and the geometry of the track doesn't care. Arc length is the one parametrization that every observer agrees on, because it's measured by the curve itself: lay a tape measure along it. When we come to curvature, we'll want "how fast the direction turns" — but fast with respect to what? If we used time, a curve driven quickly would seem to bend more sharply than the same curve driven slowly, which is absurd. Measuring the turning per unit of length strips the speed out and leaves a number that belongs to the shape alone. That's why differential geometers reach for the arc-length parameter again and again: it turns messy formulas into clean ones by removing the one thing that was never geometric — the clock.

Worked example 1 — the circle

Take the circle \mathbf{r}(t) = (R\cos t,\ R\sin t). Its speed is the constant R, so the arc-length function is

s(t) = \int_0^t R\,du = R\,t \qquad\Longrightarrow\qquad t(s) = \frac{s}{R}.

Substitute back to get the unit-speed circle:

\boldsymbol{\gamma}(s) = \left(R\cos\frac{s}{R},\ R\sin\frac{s}{R}\right), \qquad \lVert \boldsymbol{\gamma}'(s)\rVert = 1.

The full circumference falls out for free: when the point returns to the start, t = 2\pi, so s = 2\pi R — the length of the circle.

Worked example 2 — the helix, and a line

The helix \mathbf{r}(t) = (\cos t,\ \sin t,\ c\,t) has constant speed \sqrt{1 + c^2}, so

s(t) = \sqrt{1 + c^2}\;t \qquad\Longrightarrow\qquad t(s) = \frac{s}{\sqrt{1 + c^2}},

and the arc-length helix is \boldsymbol{\gamma}(s) = \big(\cos\frac{s}{k},\ \sin\frac{s}{k},\ \frac{c s}{k}\big) with k = \sqrt{1 + c^2}. Whenever the speed is a constant, the reparametrization is nothing more than a rescaling of the parameter — the easy case.

A straight line makes it even plainer. Let \mathbf{r}(t) = \mathbf{p} + t\,\mathbf{v} with \lVert \mathbf{v}\rVert = L. Then s(t) = Lt, and the unit-speed line is

\boldsymbol{\gamma}(s) = \mathbf{p} + s\,\frac{\mathbf{v}}{\lVert \mathbf{v}\rVert} = \mathbf{p} + s\,\hat{\mathbf{v}},

which just walks along the line one unit of length at a time — exactly what "unit speed" should mean.

Two traps live in this section. First, arc length does not depend on the parametrization. Whether you crawl or sprint around a circle, its length is still 2\pi R; the integral \int \lVert \mathbf{r}'\rVert\,dt automatically corrects for speed, because a faster parametrization has a larger \lVert \mathbf{r}'\rVert but a proportionally shorter time interval. Length is a property of the image and orientation, not of the clock.

Second — and this is where many students come unstuck — you usually cannot solve s(t) for t in closed form. The unit-speed reparametrization always exists in principle (that's the theorem), but for most curves the integral s(t) = \int_a^t \lVert \mathbf{r}'\rVert\,du has no elementary antiderivative, and even when it does, inverting it can be impossible with formulas. The ellipse is the classic culprit: its arc-length integral is an elliptic integral, which is precisely the historical example that has no closed form. So arc-length parametrization is a superb theoretical tool — we prove things with it — but in practice you rarely write \boldsymbol{\gamma}(s) down explicitly. The circle, helix and line are lucky exceptions because their speed is constant.

Speed builds up into length

For the ellipse \mathbf{r}(t) = (2\cos t,\ \sin t) the speed \lVert \mathbf{r}'(t)\rVert = \sqrt{4\sin^2 t + \cos^2 t} is not constant — it pulses up at the flat ends of the ellipse and drops at the pointy ones. The lower curve is that speed; the upper curve is the running arc length s(t) = \int_0^t \lVert \mathbf{r}'\rVert\,du. Wherever the speed is high the length climbs steeply; the slope of the length curve is exactly the speed below it, because ds/dt = \lVert \mathbf{r}'\rVert. The total height reached at t = 2\pi is the perimeter of the ellipse — a number with no elementary closed form.