Arc length

The arc length of a smooth curve $\mathbf{r}(t)$ from $t=a$ to $t=b$ is

$L={\int }_a^b|{\mathbf{r}}^{\prime }(t)|dt.$

In words: arc length is the integral of the speed. Chop the parameter interval into small subintervals; each gives a chord of approximate length $|{\mathbf{r}}^{\prime }(t)|dt$. Sum and take the limit.

Derivation

For small $\Delta t$, the chord from $\mathbf{r}(t)$ to $\mathbf{r}(t+\Delta t)$ has length

$|\mathbf{r}(t+\Delta t)-\mathbf{r}(t)|\approx |{\mathbf{r}}^{\prime }(t)|\Delta t.$

Sum over a partition of $[a,b]$, refine, and take the limit. The Riemann sum becomes the integral ${\int }_a^b|{\mathbf{r}}^{\prime }(t)|dt$, provided $|{\mathbf{r}}^{\prime }(t)|$ is continuous (which holds for smooth curves).

Examples

Helix $\mathbf{r}(t)=⟨\cos t,\sin t,t⟩$ from $t=0$ to $t=2\pi$.

Speed $|{\mathbf{r}}^{\prime }|=\sqrt{2}$ (constant), so $L={\int }_0^{2\pi }\sqrt{2}dt=2\pi \sqrt{2}$.

Sanity check: unwound onto a plane, the helix becomes a right triangle with legs $2\pi$ (one revolution of the circle, horizontal axis) and $2\pi$ ($z$-rise). Hypotenuse $\sqrt{(2\pi {)}^2+(2\pi {)}^2}=2\pi \sqrt{2}$. ✓

Straight line $\mathbf{r}(t)=⟨4t,3t⟩$ from $t=0$ to $t=2$.

$|{\mathbf{r}}^{\prime }|=5$, so $L=10$. Direct check: $\mathbf{r}$ goes from origin to $(8,6)$, distance $\sqrt{64+36}=10$. ✓

Arc-length parameterization

The function

$s(t)={\int }_a^t|{\mathbf{r}}^{\prime }(\tau )|d\tau$

is strictly increasing (when ${\mathbf{r}}^{\prime }\ne 0$) and so has an inverse $t(s)$. Reparameterizing by $s$ (the arc length) gives a curve with $|{\mathbf{r}}^{\prime }(s)|=1$ — speed identically $1$.

We rarely reparameterize explicitly in practice — the integral defining $s(t)$ is usually nasty. But arc-length parameterization is the canonical choice in differential geometry: it makes the parameter a geometric invariant (length along the curve) rather than an arbitrary label.

In arc-length parameterization, the unit tangent equals the derivative: $\mathbf{T}(s)={\mathbf{r}}^{\prime }(s)$.

Why this matters

Arc length is the conversion factor between the parameter $t$ and the geometric length $s$:

$ds=|{\mathbf{r}}^{\prime }(t)|dt.$

This $ds$ appears in scalar line integrals ${\int }_{C}fds$, which integrate a scalar function along a curve. The $|{\mathbf{r}}^{\prime }(t)|$ factor is what makes the line integral parameterization-independent.

The vector line integral uses $d\mathbf{r}={\mathbf{r}}^{\prime }(t)dt$ instead — direction information is kept, not just magnitude.