 Geodesics of two-dimensional de Sitter spacetime

A geodesic on the embedded two-dimensional de Sitter spacetime is the intersection with a two-plane through the ambient origin. For initial point $P$ and tangent $V$ satisfying $P^2=-1$, $P\cdot V=0$, $V^2=\epsilon$, its affine parametrization is $P\cosh s+V\sinh s$ when timelike, $P+sV$ when null and $P\cos s+V\sin s$ when spacelike. These solve $X^{\prime \prime }=\epsilon X$. The spacelike geodesics are closed; timelike geodesics and null geodesics extend to both conformal infinities with infinite affine length.