Hyperbolic space.

Returns the new_manifold object describing the hyperbolic space \(H^d\) of constant curvature \(-1\), in the hyperboloid model $$H^d = \{x \in R^{d+1} : -x_0^2 + x_1^2 + \dots + x_d^2 = -1,\ x_0 > 0\}.$$ Points are represented as rows \((x_0, x_1, \ldots, x_d)\), so data on \(H^d\) have d + 1 columns. This is the object used by manifold.dcov.test and choose_base_point when manifold = "hyperbolic".

Usage

manifold_hyperbolic()

Value

An object of class "manifold".

Details

The exponential map at any point is a global diffeomorphism, so the cut locus is empty and every base point is admissible. Its components are is_in_hyperbolic, log_map_hyperbolic, dist_hyperbolic, is_valid_hyperbolic, and two base point methods: "auto" (the default, the closed-form Lorentzian centroid auto_base_point_hyperbolic) and "optimize" (optimize_base_point_hyperbolic).

Examples

manifold_hyperbolic()
#> <manifold: hyperbolic>
#>   representation:     the hyperboloid model {x in R^(d+1): -x_0^2 + x_1^2 + ... + x_d^2 = -1, x_0 > 0}
#>   base point methods: auto, optimize