$PSL(2,\mathbb{R})$ and the set of oriented geodesics

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enter image description here How can $G/D$ be identified with the set of (oriented) geodesics?

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$G$ acts transitively on the oriented geodesics and the subgroup fixing the oriented imaginary axis (ie. fixing both $i\infty$ and $0$) is $D$.

The oritended geodesic represented by $gD$ is $gD0\to gDi\infty$.