Arc Length

Let $\boldsymbol{r}(t)$ be defined on $[a, b]$, and let $P = \{t_0, t_1, \dots, t_n\}$, $a = t_0 < t_1 < \dots < t_n = b$, be a partition of $[a, b]$. The polygonal length corresponding to $P$ is

The curve is rectifiable iff these polygonal lengths have a finite least upper bound, which is the arc length of the curve.

Examples:

• $\boldsymbol{r}(t) = a\cos t\,\boldsymbol{i} + a\sin t\,\boldsymbol{j}$, $0 \le t \le 2\pi$, is rectifiable: $\boldsymbol{r}'(t)$ is continuous, so the arc length integral applies directly, giving $L = 2\pi a$.
• $f(x) = x\sin(1/x)$ for $0 < x \le 1$, $f(0) = 0$, is continuous but not rectifiable: near $x = 0$ the graph oscillates infinitely often, and the amplitude does not decay fast enough to keep the polygonal lengths bounded.

If $x = x(t)$, $y = y(t)$, $z = z(t)$ have continuous derivatives on $[a, b]$, the arc length from $t = a$ to $t = b$ is

Equivalently, for $C$ the curve traced by continuously differentiable $\boldsymbol{r}(t)$,

Change of Parameter

A change of parameter in $\boldsymbol{r}(t)$ is a substitution $t = g(\tau)$ producing $\boldsymbol{r}(g(\tau))$, tracing the same curve as $\boldsymbol{r}(t)$, possibly in a different direction as $\tau$ increases.

If $\boldsymbol{r}(t)$ is differentiable with respect to $t$, and $t = g(\tau)$ is a change of parameter with $g$ differentiable with respect to $\tau$, then $\boldsymbol{r}(g(\tau))$ is differentiable with respect to $\tau$, and

Equivalently,

Arc-Length Parametrization

Let $C$ be the curve traced by a continuously differentiable $\boldsymbol{r}(t)$ with $\boldsymbol{r}'(t) \ne \vec{0}$, and let $\boldsymbol{r}(t_0)$ be a reference point on $C$. The signed arc-length parameter $s$ measured from $\boldsymbol{r}(t_0)$ is

This is an odometer reading, integrating speed over time gives distance travelled.

Since $\text{d}s/\text{d}t = \lVert \boldsymbol{r}'(t) \rVert > 0$, $s = s(t)$ is strictly increasing, hence invertible into $\boldsymbol{r}(s)$. A parametrization expressed in terms of $s$ is an arc-length parametrization.

Properties

$C$ is the curve traced by a continuously differentiable $\boldsymbol{r}(t)$ with $\boldsymbol{r}'(t) \ne \vec{0}$, and $s$ is the signed arc-length parameter defined above.

$s = s(t)$ is the arc-length parameter for $C$ iff

This is the defining integral restated as a derivative: $s$ advances at exactly the rate $C$ is traced, not some rate tied to an arbitrary parameter $t$.

If $\boldsymbol{r}$ is expressed as a function of $s$ instead of $t$, then

everywhere, almost by definition: since $s$ counts distance travelled, advancing it by one unit moves $\boldsymbol{r}(s)$ exactly one unit along $C$. This is why an arc-length parametrization is aka. a unit-speed parametrization.

Conversely, if $\lVert \text{d}\boldsymbol{r}/\text{d}t \rVert = 1$ for all $t$, then for any $t_0$ in the domain of $\boldsymbol{r}$,

is already an arc-length parameter with reference point $\boldsymbol{r}(t_0)$: a unit-speed parametrization needs no reparametrization through the integral formula, only a shift so that $s = 0$ at the reference point.

Examples:

• For $x^2 + y^2 = a^2$, $a > 0$, with counterclockwise orientation and reference point $(a, 0)$: $\boldsymbol{r}(t) = a\cos t\,\boldsymbol{i} + a\sin t\,\boldsymbol{j}$, with $t_0 = 0$ giving $\boldsymbol{r}(t_0) = (a, 0)$. Since $\lVert \boldsymbol{r}'(t) \rVert = a$ is constant, $s = at$, so the arc-length parametrization is $\boldsymbol{r}(s) = a\cos(s/a)\,\boldsymbol{i} + a\sin(s/a)\,\boldsymbol{j}$.

Worked Example

Find the arc length of $\boldsymbol{r}(t) = 3\cos t\,\boldsymbol{i} + 3\sin t\,\boldsymbol{j} + 4t\,\boldsymbol{k}$ for $0 \le t \le 2\pi$, with reference point $\boldsymbol{r}(0) = (3, 0, 0)$. Then find its arc-length parametrization.

Differentiate each component.

Compute the speed.

The derivatives are continuous and $\boldsymbol{r}'(t) \ne \vec{0}$, so the formula applies.

Compute the arc-length parameter with $t_0 = 0$.

Invert to get $t = s/5$ and substitute into $\boldsymbol{r}$.

Verify the unit speed.

The arc length is $L = 10\pi$, and the arc-length parametrization is $\boldsymbol{r}(s)$ for $0 \le s \le 10\pi$.