Recipe

Find the arc length of a space curve and reparametrize by arc length

Use when: you have r(t) and need either its length over an interval, or the same curve rewritten so the parameter itself measures distance traveled

Goal: the arc-length function s(t) and, when it inverts in closed form, the reparametrized curve r(s)

Taught in The unit tangent and arc-length parametrization

Quick recipe

Differentiate $\mathbf r(t)$ and find the speed $|\mathbf r'(t)|$
Integrate speed to get the arc-length function $s(t)$
Solve $s(t)$ for $t$ , the inversion step
Substitute $t(s)$ back into $\mathbf r(t)$

1
Differentiate each component, then take the magnitude of the result.

Example

$\mathbf r(t)=\langle5\cos t,5\sin t,12t\rangle$ : $\mathbf r'(t)=\langle-5\sin t,5\cos t,12\rangle$ , and $|\mathbf r'(t)|=\sqrt{25\sin^2t+25\cos^2t+144}=\sqrt{169}=13$ , constant for every $t$ .

Why: arc length accumulates speed over time, so both this step's result and the reparametrization steps ahead start from this one magnitude.
2
Set up $s(t)=\displaystyle\int_a^t|\mathbf r'(u)|\,du$ , starting from whatever value of $t$ counts as the beginning, and evaluate it.

Example

$s(t) = \displaystyle\int_0^t 13\,du = 13t$ .

⚡ Shortcut

When speed is constant, as it is here, skip setting up the integral entirely. Arc length is just speed times elapsed time, $s(t)=|\mathbf r'|\cdot t$ .

Why: this is the same "distance equals the integral of speed" idea from single-variable motion, with $|\mathbf r'(t)|$ standing in for the speedometer reading.
3
Rearrange $s=s(t)$ to isolate $t$ as a function of $s$ .

Example

$s=13t\ \Longrightarrow\ t=s/13$ .
- speed is constant — inversion is immediate, $t=s/|\mathbf r'|$ .
- speed is not constant — $s(t)$ usually has no closed-form inverse, and most textbook problems are built with constant speed for exactly this reason.
Why: reparametrizing by arc length means relabeling every point on the curve by its own distance traveled instead of the original parameter, and that requires running $s(t)$ backward.
4
Replace every $t$ in the original $\mathbf r(t)$ with the expression for $t$ found in Step 3.

Example

$\mathbf r(s) = \left\langle5\cos\dfrac{s}{13},\,5\sin\dfrac{s}{13},\,\dfrac{12s}{13}\right\rangle$ .

Why: this is the arc-length parametrization itself, the same curve now indexed by distance traveled rather than by time.

Common mistakes

Forgetting the starting point $a$ in $s(t)=\int_a^t|\mathbf r'(u)|\,du$ , which measures arc length from the wrong place.
Substituting $t(s)$ into $\mathbf r'(t)$ instead of into $\mathbf r(t)$ itself.
Assuming every curve reparametrizes in closed form, when most curves' $s(t)$ can't be inverted by hand at all.
Stopping at $s(t)$ and reporting it as the final answer instead of finishing the substitution back into $\mathbf r(t)$ .