Length of a curve ($L(\alpha )$)

The length of a piecewise smooth curve is its arc length: $L(\alpha )=\int \Vert {\alpha }^{\prime }(t)\Vert dt$, 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 $\Vert q-p\Vert$, while a route that bends or retraces can be longer. On a Riemannian manifold, comparing these lengths over all routes defines Riemannian distance.