Arc length

The idea is to approximate the curve by a polygonal path: divide \([a,b]\) into small steps \(\Delta t\), each piece has length about \(|\mathbf r'(t)|\,\Delta t\) (speed times time), and summing and taking the limit gives the integral of the speed \(|\mathbf r'(t)|\). In the plane, with \(\mathbf r(t)=\langle x(t),y(t)\rangle\), the integral is written \(\int_a^b\sqrt{x'(t)^2+y'(t)^2}\,dt\), and in space \(z'(t)^2\) is added under the root.

For example, for the circle \(\langle R\cos t,R\sin t\rangle\) the speed is constant, \(|\mathbf r'|=R\), and the length over \([0,2\pi]\) is \(2\pi R\), the familiar perimeter. The helix \(\langle\cos t,\sin t,t\rangle\) has speed \(\sqrt2\), so one turn has length \(2\sqrt2\,\pi\), more than the base circle because the curve also climbs. When the speed is not constant the integral may have no elementary antiderivative, and it is then computed numerically.

Length depends on the curve and on how many times it is traversed, not on how fast it is traversed: \(\langle\cos2t,\sin2t\rangle\) on \([0,\pi]\) has the same length \(2\pi\) as \(\langle\cos t,\sin t\rangle\) on \([0,2\pi]\). For this to hold one must watch out for repeated pieces: if the curve passes twice over the same arc, that portion counts twice.

The arc length function \(s(t)=\int_a^t|\mathbf r'(u)|\,du\) measures the distance covered from the start up to time \(t\). Reparametrizing by \(s\) gives a curve of unit speed, useful later to define curvature independently of the parametrization.