Projections of surfaces in the hyperbolic space along horocycles

We study in this paper orthogonal projections of embedded surfaces $M$ in $H^3_+(-1)$ along horocycles to planes. The singularities of the projections capture the extrinsic geometry of $M$ related to the lightcone Gauss map. We give geometric characterisations of these singularities and prove a Koenderink type theorem which relates the hyperbolic curvature of the surface to the curvature of the profile and of the normal section of the surface. We also prove duality results concerning the bifurcation set of the family of projections