Length of a curve
= Length of a curve
{title2=$L(\alpha)$}
The length of a piecewise smooth <curve> is its <arc length>: $L(\alpha)=\int\|\alpha'(t)\|\,dt$, using either the <Euclidean space> <norm> or the <norm> from a specified <Riemannian metric>. A straight segment from $p$ to $q$ in <Euclidean space> has <length> $\|q-p\|$, while a route that bends or retraces can be longer. On a <Riemannian manifold>, comparing these <lengths> over all routes defines <Riemannian distance>.