Arc Length & Reparametrization

Measure curves by arc length so geometry is intrinsic and unit-speed.

The idea

Arc length measures distance along a curve, and reparametrizing by arc length removes the tracing speed from a curve's description, leaving only its shape.

A parametrization $\alpha(t)$ records two different things: the shape the curve draws in space and the schedule on which it draws it. The velocity $\alpha'(t)$ gives the direction of motion, and its norm $\lVert \alpha'(t) \rVert$ is the speed. Integrating the speed gives the arc length from $a$ to $t$: $s(t) = \int_{a}^{t} \lVert \alpha'(u) \rVert \, du.$ A reparametrization retraces the same curve on a different schedule. It changes the speed at every instant but leaves the arc length invariant: length belongs to the curve, speed to the tracing.

A curve is unit-speed, or parametrized by arc length, when $\lVert \alpha'(t) \rVert = 1$ everywhere; then $s(t) = t - a$ and the parameter itself measures distance travelled. Every regular curve admits such a parametrization: compute $s(t)$, invert it, and substitute. Once the speed is fixed at $1$, whatever still varies along the curve is a property of the curve itself, not of the parametrization we chose.

Ways to work on it

• Walkthrough. Speed, arc length, and what unit-speed means.
• Practice. Compute the arc length of a constant-speed curve.
• Hardest. Reparametrize a non-constant-speed curve by arc length.

Not sure where to start? Take the ten-question placement test.