Stereographic geometry of coherence and which-path information

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.