= Solution
An <embedded surface parametrization> is a smooth map $F:U\subset\mathbb R^2\to\mathbb R^3$, with open $U$, injective differential of rank two at each point, and $F$ a <homeomorphism> onto its image with the subspace topology. Its <induced metric> is
$$
g_{ij}=\partial_iF\cdot\partial_jF,\qquad ds^2=g_{ij}du^idu^j.
$$
In coordinates, an arc-length-parametrized <geodesic> satisfies
$$
\boxed{\ddot u^k+\Gamma^k_{ij}\dot u^i\dot u^j=0,\qquad
g_{ij}\dot u^i\dot u^j=1,}
$$
where the <Christoffel symbols> are $\Gamma^k_{ij}=\tfrac12g^{k\ell}(\partial_i g_{j\ell}+\partial_j g_{i\ell}-\partial_\ell g_{ij})$. The normalization fixes unit speed; it is compatible with the differential equations because geodesic speed is constant.
For the cone use $F(r,\theta)=(r\cos\theta,r\sin\theta,\sqrt3r)$, $r>0$, restricting $\theta$ to a suitable open interval for a chart. Direct differentiation gives its <first fundamental form>
$$
ds^2=4\,dr^2+r^2d\theta^2.
$$
With $\rho=2r$ and $\phi=\theta/2$, this becomes $d\rho^2+\rho^2d\phi^2$, the Euclidean <metric> in <polar coordinates>. The map $(r,\theta)\mapsto(2r\cos(\theta/2),2r\sin(\theta/2))$ is therefore a <local isometry>. It is local rather than global because increasing $\theta$ by $2\pi$ increases $\phi$ by $\pi$.
Given two points, choose their angle representatives so that $|\theta_2-\theta_1|\leq\pi$. Their developed plane angles then differ by at most $\pi/2$. The straight segment between their developed positions stays away from the origin: it lies in an angular sector of width at most $\pi/2$ containing both positive-radius endpoints. Develop that segment back onto the cone. Since <local isometries> preserve <geodesics>, it gives a geodesic joining the two points; it can be parametrized by arc length. This constructs <geodesics on a punctured circular cone>. For a repeated point the constant curve is the degenerate geodesic. In particular, this argument does not appeal to completeness of the punctured cone.
\b[The joining geodesic need not be unique.] For endpoints with equal positive radius and opposite azimuth, one may use the angular changes $+\pi$ and $-\pi$. In the plane these give two straight segments from $(\rho,0)$ to $(0,\rho)$ and $(0,-\rho)$, respectively. Both avoid the origin. Their lifts pass on opposite sides of the cone and have distinct images, so a <reparametrization> cannot identify them.
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