TITLE: Coordinates and distance in higher dimensional spherical and hyperbolic space
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: There's nothing quite as simple as your example for hyperbolic space, because the example is based on embedding the $n$-sphere in Euclidean $n+1$-space, and no such smooth isometric embedding is possible for hyperbolic space.
The closest to what you're asking might be to use $n$ coordinates $x, y, \ldots, z$ with a metric given by
$$d\sigma^2 = \frac{dx^2+dy^2+\cdots+dz^2}{(1+\frac{k}{4}(x^2+y^2+\cdots+z^2))^2}$$
where varying $k$ can produce either Euclidean space for $k=0$, an $n$-sphere minus one point (via stereographic projection) for $k>0$, and hyperbolic space for $k<0$ and $x^2+y^2+\cdots+z^2<4/k$ (which produces the $n$-dimensional analogue of the Poincaré disk model). In each cases, the denominator depends only on which point we're at, and the tangent space at any point is a simple uniform scaling away from the coordinate tangent space.