Arc Length and Reparametrisation: Explained with Examples

The arc length of a curve is the total distance travelled along it, like a car's odometer reading after a winding drive. For a parametrised curve r(t) = (x(t), y(t), z(t)) with t running from a to b, the arc length is L = the integral from a to b of |r'(t)| dt, which adds up infinitesimal straight-line pieces.

Arc length captures the true geometric length of a path, independent of how the path is traced. A circle drawn quickly and the same circle drawn slowly have identical arc lengths. Engineers use it for cable and rope lengths, computer graphics uses it to space objects evenly along a path, and physicists use it to describe motion in the cleanest way.

Key Concepts

• Parametrised curve: a map r from [a, b] into space; the parameter t acts like a clock carried along the curve.
• Speed: |r'(t)|, the rate of travel per unit of the parameter t.
• Regular curve: r'(t) is never the zero vector, so a tangent direction exists everywhere.
• Arc-length function: s(t) = integral from a to t of |r'(u)| du, the distance travelled up to parameter t.
• Reparametrisation: a smooth change of parameter, gamma = r composed with h; the set of points is unchanged.

Formulas

• Arc length: L = integral from a to b of |r'(t)| dt. Split the integral at any sharp corners.
• Arc-length parameter: s(t) = integral from a to t of |r'(u)| du, with ds/dt = |r'(t)|.
• Unit-speed form: with t(s) the inverse of s(t), gamma(s) = r(t(s)) satisfies |gamma'(s)| = 1 everywhere.

Worked Example

Find the length of one turn of the helix r(t) = (cos t, sin t, t), t from 0 to 2 pi, and reparametrise it by arc length.

1. r'(t) = (-sin t, cos t, 1), so |r'(t)| = sqrt(sin^2 t + cos^2 t + 1) = sqrt(2).
2. L = integral from 0 to 2 pi of sqrt(2) dt = 2 pi sqrt(2), about 8.886.
3. s(t) = t sqrt(2), hence t = s / sqrt(2), with s from 0 to 2 pi sqrt(2).
4. gamma(s) = (cos(s/sqrt(2)), sin(s/sqrt(2)), s/sqrt(2)).
5. gamma'(s) has length sqrt(1/2 + 1/2) = 1, confirming unit speed.

Common Misconceptions

• Arc length is not the endpoint distance. The helix length 8.886 exceeds |r(2 pi) - r(0)| = 2 pi ≈ 6.283, because the helix coils as it rises.
• Reparametrising never changes the curve. Only the clock changes; the geometric coil is identical.
• s(t) needs nonzero speed. A path that stops and reverses cannot be smoothly reparametrised by arc length at the stop.

Real-World Applications

• GPS navigation: trip distance integrates speed along the road, not straight-line distance.
• Manufacturing: arc-length parametrisation keeps CNC cutting speed constant along tool paths.
• Animation: constant visual speed along a curve requires the arc-length parameter.

Arc length is the integral of speed along a curve: its true geometric length. Reparametrisation changes only the parameter, and the arc-length parameter turns every regular curve into a unit-speed one, the standard starting point for studying curvature.