Lorentz Logarithmic Map

Map points on the Lorentz hyperboloid to the tangent space at mu.

Usage

lorentz_log_map(Z, mu = NULL, tol = 1e-10)

Arguments

Z

n x 3 Lorentz hyperboloid coordinates.

mu

Reference point on the Lorentz hyperboloid. If NULL, the normalized extrinsic mean is used.

tol

Intrinsic-distance threshold for setting a log-map row to zero; the default is 1e-10 in unit-curvature distance units.

Value

An n x 3 matrix of tangent vectors in T_mu H^2.

Details

For unit-hyperboloid points \(z\) and \(\mu\), define \(a=-\langle\mu,z\rangle_L\) and the intrinsic distance \(d=\operatorname{acosh}(a)\) using lorentz_inner() and hyperbolic_distance(). The logarithmic map is $$\log_\mu(z)=\frac{d}{\sinh(d)}(z-a\mu),\qquad d>0.$$ The result is a tangent vector orthogonal to \(\mu\) under the Lorentz inner product. At \(z=\mu\) the continuous extension is the zero vector. Numerically, tol is compared with the distance \(d\), not with \(a-1\): rows with \(d\leq\mathrm{tol}\) are set to zero. Other rows use stable, algebraically equivalent coordinate formulas.

Examples

H <- simulate_H2(5, r_max = 2)
V <- lorentz_log_map(H$Z)
dim(V)