Special Euclidean Group

The group of rigid-body transformations in dimensions: combines a rotation and a translation into a single manifold element.

Geometry Summary

The weight field controls the relative importance of rotation vs. translation in the metric:

• weight = 1.0: equal weighting (default)
• weight >> 1: translation-dominant (large-scale SLAM)
• weight << 1: rotation-dominant (attitude estimation)

Note: does not implement Curvature. Homogeneous matrix representation $(N+1)\times(N+1)$ is unavailable due to the unstable generic_const_exprs feature; rotation and translation are stored separately in SEPoint.

Formula Reference

Group operation and inverse:

Exponential map (via exp and left Jacobian ):

where and are body-frame components.

Logarithmic map:

Riemannian metric:

where and .

Code Examples

Constructing

use cartan::prelude::*;
use cartan::manifolds::{SpecialEuclidean, SEPoint};
use nalgebra::{SMatrix, SVector};
 
// SE(3) with default (equal) weighting
let se3 = SpecialEuclidean::<3> { weight: 1.0 };
 
// SE(3) with translation-dominant metric (useful for large-scale SLAM)
let se3_trans = SpecialEuclidean::<3> { weight: 10.0 };
 
let mut rng = rand::rng();
let p = se3.random_point(&mut rng);   // random rigid-body transform
let v = se3.random_tangent(&p, &mut rng);

Exp/log roundtrip

let q = se3.exp(&p, &v);
let v_back = se3.log(&p, &q).unwrap();
 
// Rotation and translation components recover separately
assert!((v_back.rotation - v.rotation).norm() < 1e-10);
assert!((v_back.translation - v.translation).norm() < 1e-10);

Group composition

// Compose two rigid-body transforms manually
let r1 = p.rotation;
let t1 = p.translation;
let r2 = q.rotation;
let t2 = q.translation;
 
let composed = SEPoint {
    rotation: r1 * r2,
    translation: r1 * t2 + t1,
};

Geodesic interpolation

let mid = se3.geodesic(&p, &q, 0.5).unwrap();
// mid is the midpoint rigid transform on the geodesic from p to q

Numerical Notes

Injectivity radius: π; the logarithm fails when the rotational component is a rotation (i.e., has eigenvalue $-1$). Returns Err at the cut locus of .

Weight selection: The weight does not affect whether exp/log succeed, only the geodesic path. For optimization tasks, tuning weight to match the physical scale of rotations vs. translations improves convergence.

No Curvature impl: is not flat (the factor has positive curvature) but Curvature is not yet implemented. Use cartan-geo's CurvatureQuery on a SpecialOrthogonal<N> manifold for rotation-only curvature.

Applications

• Robotics: for 6-DOF end-effector poses, for mobile robot configurations
• SLAM: Lie group optimization of robot trajectories
• Computer vision: camera pose estimation, structure-from-motion
• Motion planning: interpolating between rigid-body configurations
• Biomechanics: modeling joint kinematics in N-dimensional configuration space