  Recently, it was shown that quantum entanglement is an indispensable part of
the duality behavior of light. Here, we report a surprisingly intimate
connection between the stereographic projection and the duality--entanglement
nature of a single photon. We show that, the duality--entanglement relation
[Optica \textbf{5}, 942 (2018)], naturally emerges from the stereographic
projection geometry. We demonstrate that this geometry is complementarity
sensitive; in the sense that, it is sensitive to the particle nature, wave
nature, and entanglement nature of a single photon.
