As nouns the difference between hyperplane and horosphere

is that hyperplane is (geometry) an n''-dimensional generalization of a plane; an affine subspace of dimension ''n-1'' that splits an ''n -dimensional space (in a one-dimensional space, it is a point; in two-dimensional space it is a line; in three-dimensional space, it is an ordinary plane) while horosphere is an $n-1$-dimensional hyperplane in hyperbolic $n$-dimensional space: it is (in the ) euclidean-tangent at infinity to the boundary sphere or (in the upper-half-space model) euclidean-parallel to the boundary hyperplane.