= Geodesics on a punctured circular cone
{title2=$ds^2=d\rho^2+\lambda^2\rho^2d\theta^2$}
For a circular cone with intrinsic metric $d\rho^2+\lambda^2\rho^2d\theta^2$, $0<\lambda\leq1$, local development uses plane angle $\phi=\lambda\theta$. A plane straight segment that avoids the origin lifts to a <geodesic>. Choosing an angular difference at most $\pi$ gives plane difference at most $\pi\lambda$, so for $\lambda<1$ this constructs a joining geodesic between any two distinct points. Different permitted angle lifts can yield distinct joining geodesics; the missing apex cannot be used to join straight segments through the plane origin.
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