= Line element
{title2=$ds^2=g_{ij}dx^idx^j$}
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 $ds^2=R^2d\theta^2+dz^2$. Unrolling with $u=R\theta$ yields the Euclidean form $du^2+dz^2$, explaining the straight unrolled descriptions of <geodesics on a circular cylinder>.
Back to article page
