Geometry

Riemannian manifolds, Lie groups, differential geometry, projective geometry, fiber bundles, ODE integrators, and tensors: all over the binary fixed-point types.

What it is

The geometric layer builds on FixedVector/FixedMatrix and the linear algebra modules to provide coordinate-free constructs: geodesics and parallel transport on manifolds, exp/log and the adjoint on Lie groups, curvature tensors, projective transforms, connections on bundles, ODE integration, and arbitrary-rank tensors. Closed-form algorithms are used where they exist (Rodrigues for SO(3), the V-matrix for SE(3)); everything else composes the wide-tier matrix functions.

Usage

use g_math::fixed_point::{FixedVector, FixedPoint};
use g_math::fixed_point::imperative::manifold::{Manifold, Sphere};

let s = Sphere { dim: 2 };                    // unit 2-sphere (ambient = dim + 1)
let p = FixedVector::from_slice(&[FixedPoint::one(), FixedPoint::ZERO, FixedPoint::ZERO]);
let q = FixedVector::from_slice(&[FixedPoint::ZERO, FixedPoint::one(), FixedPoint::ZERO]);
let d = s.distance(&p, &q).unwrap();          // geodesic distance
let v = s.log_map(&p, &q).unwrap();           // tangent vector p → q
let back = s.exp_map(&p, &v).unwrap();        // ≈ q

What's here

Manifolds (manifold): trait exp_map, log_map, distance, parallel_transport, inner_product: Euclidean, Sphere, Hyperbolic (hyperboloid model), SPD, Grassmannian, Stiefel, products.
Lie groups (lie_group): trait adds lie_exp, lie_log, hat/vee, adjoint, bracket, act: SO(3) via closed-form Rodrigues, SE(3) via closed-form V-matrix, plus SO(n), GL(n), O(n), SL(n) via matrix exp/log.
Differential geometry (curvature): Christoffel symbols, Riemann/Ricci/scalar/sectional curvature, geodesic integration, parallel transport along curves.
Projective (projective): homogeneous coordinates, projective transforms, cross-ratios, stereographic projection, Möbius transformations (real and complex).
Fiber bundles (fiber_bundle): trivial, vector (connection coefficients, horizontal lift, parallel transport, curvature 2-form), principal (transition cocycles).
ODE solvers (ode): RK4 (fixed step), Dormand-Prince RK45 (adaptive), symplectic Störmer-Verlet (energy-preserving). Weighted sums accumulate at the compute tier; step halving is an exact bit shift.
Tensors (tensor, tensor_decompose): arbitrary-rank contraction, outer product, trace, index raising/lowering via a metric, (anti)symmetrization; decompositions truncated_svd, tucker_decompose (HOSVD), cp_decompose (ALS).

Public API

See PUBLIC_API.md → Geometry, → ODE, → Tensors, and docs.rs.

Behaviour & limits

All operations are in the binary domain via FixedMatrix/FixedVector.
Manifold maps (Sphere, hyperbolic, SPD, Grassmannian, Stiefel, products), Lie-group maps (SO(3), SE(3), SO(n), GL(n), SL(n)), fiber-bundle transport, curvature (Christoffel symbols, Riemann, Ricci, scalar, sectional), geodesic integration, ODE integrators, projective maps and tensor decompositions carry their intermediate state at the compute tier and round once. Against mpmath (or the same scheme in exact arithmetic, for integrators and finite differences) they are within one unit on every profile and at realtime 8 to 24 fraction bits, except the SE(3) log of a rounded exp near pi (3 units) and CP-ALS reconstruction (2). Gates: tests/manifold_compute_tier_validation.rs, tests/lie_fiber_compute_tier_validation.rs, tests/curvature_compute_tier_validation.rs, tests/ode_compute_state_validation.rs, tests/projective_tensor_compute_tier_validation.rs.
Angles are formed from exact products (atan2 of the sine and cosine parts, the logarithmic form of acosh), so close points keep their precision. Accuracy of formula-sensitive constructs still depends on input representability (see the precision guide).
HyperbolicSpace distance and log return Err(DomainError) for points off the upper sheet; SPDManifold::inner_product panics on a singular base point.

Determinism guarantees are in CONTRACT.md.

Disclaimer

This software is provided "as is", without warranty of any kind, express or implied. Use of this software is entirely at your own risk. In no event shall the author or contributors be held liable for any damages arising from the use or inability to use this software.

Built by Niels Erik Toren · support & donations.

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