Then is an isometry mapping to and a point on the positive x-axis to . is a translation through the line of distance . If , it is in the direction. If , it is in the direction. is the group of translations through . Symmetries of horospheres Let H be some horosphere such that points of the form are inside of it for arbitrarily large x. For any vector b in is a hororotation that maps H to itself. The set of such hororotations is the group of hororotations preserving H. All hororotations are conjugate to each other. For any in O(n-1) is a rotation or reflection that preserves H and the x-axis. These hororotations, rotations, and reflections generate the group of symmetries of H. The symmetry group of any horosphere is conjugate to it. They are isomorphic to the Euclidean group E(n-1). History In several papers between 1878-1885, Wilhelm Killing used the representation he attributed to Karl Weierstrass for Lobachevskian geometry. In particular, he discussed quadratic forms such as or in arbitrary dimensions , where is the reciprocal measure of curvature, denotes Euclidean geometry, elliptic geometry, and hyperbolic geometry. According to Jeremy Gray (1986), Poincaré used the hyperboloid model in his personal notes in 1880. Poincaré published his results in 1881, in which he discussed the invariance of the quadratic form . Gray shows where the hyperboloid model is implicit in later writing by Poincaré. Also Homersham Cox in 1882 used Weierstrass coordinates (without using this name) satisfying the relation as well as . Further exposure of the model was given by Alfred Clebsch and Ferdinand Lindemann in 1891 discussing the relation and . Weierstrass coordinates were also used by Gérard (1892), Felix Hausdorff (1899), Frederick S. Woods (1903)], Heinrich Liebmann (1905). The hyperboloid was explored as a metric space by Alexander Macfarlane in his Papers in Space Analysis (1894). He noted that points on the hyperboloid could be written as where α is a basis vector orthogonal to the hyperboloid axis. For example, he obtained the hyperbolic law of cosines through use of his Algebra of Physics. H. Jansen made the hyperboloid model the explicit focus of his 1909 paper "Representation of hyperbolic geometry on a two sheeted hyperboloid". In 1993 W.F. Reynolds recounted some of the early history of the model in his article in the American Mathematical Monthly. Being a commonplace model by the twentieth