 Line element

The line element of a Riemannian metric is its local expression for squared infinitesimal length. Along a parametrized curve $x(t)$, the arc length is $\int \sqrt{g_{ij}(x(t)){\dot{x}}^i{\dot{x}}^j}dt$. For a circular cylinder of radius $R$ parametrized by angle $\theta$ and axial position $z$, the induced metric gives $d{s}^2=R^{2}d{\theta }^2+d{z}^2$. Unrolling with $u=R\theta$ yields the Euclidean form $d{u}^2+d{z}^2$, explaining the straight unrolled descriptions of geodesics on a circular cylinder.