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We consider the contact between curves and hyperhorospheres in hyperbolic 4-space as an application of the theory of singularities of functions. We define the osculating hyperhorosphere and the horospherical hypersurface of the curve whose singular points correspond to the locus of polar vectors of osculating hyperhorospheres of the curve. One of the main results is a generic classification of singularities of horospherical hypersurfaces of curves.
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