= Arc length
{title2=$L(\gamma)$}
{wiki}
= Path length
{synonym}
= Arclength
{synonym}
The arc length of a piecewise smooth <curve> $\gamma:[a,b]\to M$ on a <Riemannian manifold> is the <integral> of its speed:
$$
L_g(\gamma)=\int_a^b\|\dot\gamma(t)\|_g\,dt
=\int_a^b\sqrt{g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))}\,dt.
$$
An odometer measures this accumulated travel rather than displacement between the endpoints. In <Euclidean space>, the <Riemannian metric> is the ordinary <inner product>, so the formula becomes $\int_a^b\sqrt{\sum_i\dot\gamma_i(t)^2}\,dt$.
For a <circle> of radius $R$ traversed through an <angle> $\alpha$, use $\gamma(t)=(R\cos(\alpha t),R\sin(\alpha t))$, $0\leq t\leq1$. Its speed is $R|\alpha|$, hence its <arc length> is $R|\alpha|$. Traversing a full circle twice gives $4\pi R$, even though the starting and ending points coincide. The same formula measures the two <great circle> arcs pictured in <geodesic>.
The <curve parameter> controls how fast the journey happens, not its <arc length>. A regular increasing or decreasing <reparametrization> preserves the <integral> by <change of variables>; reversing direction leaves the speed <norm> unchanged. Retracing part of a route does add <length>, because it changes the actual journey.
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