Horospherical surfaces of curves in hyperbolic space

We consider the contact between curves and horospheres in Hyperbolic 3-space as an application of singularity theory of functions. We define the osculating horosphere of the curve. We also define the horospherical surface of the curve whose singular points correspond to the locus of polar vectors of osculating horospheres of the curve. One of the main results is to give a generic classification of singularities of horospherical surface of curves