Twistor Theory of Dancing Paths

Given a path geometry on surface $\mathcal{U}$, we construct causal structure four-manifold which is the configuration space of non-incident pairs (point, path) $\mathcal{U}$. This corresponds to conformal if and only $\mathcal{U}$ real projective plane, paths are lines. We give example given by symmetric sextic, an ${\rm SL}(2,{\mathbb R})$-invariant where ellipses area $\pi$ centred at origin. shall also discuss seven-dimensional manifold corresponding conic) plane.