= Geodesics on a circular cylinder
{title2=$\theta=\theta_0+cs,\quad z=z_0+ds$}
In coordinates $(\theta,z)$ on a <circular cylinder> of radius $a$, the <induced metric> is $a^2d\theta^2+dz^2$. Its coefficients are constant, so the <Christoffel symbols> vanish and the affinely parametrized <geodesic equation> is $\theta''=z''=0$. The nonconstant solutions are generators when $c=0$, circular parallels when $d=0$, and helices when both are nonzero. Unrolling the cylinder with coordinate $u=a\theta$ makes all these curves straight lines in the Euclidean metric. Unit-speed parametrization additionally requires $a^2c^2+d^2=1$.
For a worked <helix> on a <circular cylinder> of radius $1$, choose $\theta(t)=2\pi t,z(t)=2t$ for $0\leq t\leq1$. Unrolling gives the straight segment $(u,z)=(2\pi t,2t)$, so its <arc length> is $\sqrt{4\pi^2+4}$. Its acceleration in space points normally to the surface, consistent with the <geodesic equation>.
This <geodesic> is not globally shortest between its endpoints: they share the same angular position, and the vertical generator has <length> $2$. The <helix> minimizes <length> among paths with the same endpoints and one full winding, because their unrolled lifts join $(0,0)$ to $(2\pi,2)$. In general, <geodesics> minimize sufficiently short subarcs; changing winding can produce a shorter route. Keeping this local/global distinction makes unrolling useful without mistaking every straight lift for an unrestricted shortest path.
\Image[/cylinder-unrolled-geodesic.png]
{title=A helical geodesic becomes straight when the cylinder is unrolled}
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