Struct LorentzModel 

pub struct LorentzModel<T> {
    pub c: T,
}

Lorentz (hyperboloid) model manifold.

Points live in (\mathbb{R}^{d+1}): the first component (x_0) is the “time” coordinate (always positive), the remaining (x_1, \ldots, x_d) are “space” coordinates. All points satisfy the hyperboloid constraint (\langle x, x \rangle_\mathcal{L} = -1/c).

The origin of hyperbolic space is the “north pole” ((1/\sqrt{c}, 0, \ldots, 0)).

Fields

c: T

Curvature parameter (c > 0). The hyperboloid satisfies <x,x>_L = -1/c.

Implementations

impl<T> LorentzModel<T>

where T: Float + FromPrimitive + Zero + ScalarOperand + LinalgScalar,

pub fn new(c: T) -> Self

Create a Lorentz model with curvature c (must be positive).

pub fn minkowski_dot(&self, x: &ArrayView1<'_, T>, y: &ArrayView1<'_, T>) -> T

Minkowski inner product: (\langle x, y \rangle_\mathcal{L} = -x_0 y_0 + \sum_{i=1}^d x_i y_i).

This is the indefinite bilinear form with signature ((-,+,+,\ldots,+)). For points on the hyperboloid, (\langle x, x \rangle_\mathcal{L} = -1/c).

pub fn distance(&self, x: &ArrayView1<'_, T>, y: &ArrayView1<'_, T>) -> T

Geodesic distance: (d_c(x,y) = \frac{1}{\sqrt{c}},\mathrm{arcosh}(-c\langle x,y\rangle_\mathcal{L})).

Uses a Taylor expansion (\mathrm{arcosh}(1+\delta) \approx \sqrt{2\delta}) near (\delta = 0) for numerical stability when (x \approx y).

pub fn is_on_manifold(&self, x: &ArrayView1<'_, T>, tol: T) -> bool

Check if point is on the hyperboloid: <x,x>_L = -1/c

pub fn project(&self, x: &ArrayView1<'_, T>) -> Array1<T>

Project point onto hyperboloid by scaling. Given x with x_0 > 0, find scale s such that <sx, sx>_L = -1/c.

pub fn from_euclidean(&self, v: &ArrayView1<'_, T>) -> Array1<T>

Project from Euclidean space to hyperboloid. Maps an n-dimensional Euclidean vector to the (n+1)-dimensional hyperboloid.

pub fn to_euclidean(&self, x: &ArrayView1<'_, T>) -> Array1<T>

Project from hyperboloid to Euclidean space (gnomonic projection). Maps an (n+1)-dimensional hyperboloid point to n-dimensional Euclidean.

pub fn exp_map(&self, x: &ArrayView1<'_, T>, v: &ArrayView1<'_, T>) -> Array1<T>

Exponential map at point x: maps tangent vector v to manifold. exp_x(v) = cosh(||v||_L) * x + sinh(||v||_L) * v / ||v||_L where ||v||_L = sqrt(<v,v>_L) (Lorentzian norm of tangent vector).

pub fn log_map(&self, x: &ArrayView1<'_, T>, y: &ArrayView1<'_, T>) -> Array1<T>

Logarithmic map at point x: maps manifold point y to tangent space at x. log_x(y) = d(x,y) * (y - <x,y>_L * c * x) / ||y - <x,y>_L * c * x||_L

pub fn parallel_transport( &self, x: &ArrayView1<'_, T>, y: &ArrayView1<'_, T>, v: &ArrayView1<'_, T>, ) -> Array1<T>

Parallel transport of tangent vector v from point x to point y.

pub fn origin(&self, dim: usize) -> Array1<T>

Origin of the hyperboloid: (1/sqrt(c), 0, 0, …, 0)