For n-dimensional spherical space, it seems to me the representation of points is easiest and most manipulable as unit vectors, with distance being the vector dot product (which is the cosine of the angle between the two vectors).
Is there an analogous coordinate representation for n-dimensional hyperbolic space? And with a similarly simple distance metric? I'm expecting something like an $n$- or $(n+1)$-tuple, where the distance function doesn't treat any particular function specially. In fact it would be great if there were a parameter $k$ for curvature to a distance function that would differentiate between spherical, euclidean and hyperbolic and work for any dimension. But if there are others for n-dimensional, that would be great.