Module lie 

The Lie groups SO(3) and SE(3).

Pose optimisation runs over the Lie algebra rather than over matrix entries: six free parameters instead of sixteen constrained ones, and the update T ← exp(Δξ) · T stays on the manifold by construction.

Ordering convention: ξ = [ρ; φ] — translation first, then rotation. It runs through the whole project: the same order indexes the rows of the adjoint, the columns of the residual Jacobian and the degrees of freedom in the conditioning report. It cannot be changed in one place alone.

Structs

Se3

A rigid motion: a rotation and a translation.

So3

A rotation of three-dimensional space.

Functions

absolute_orientation

The rigid motion carrying from onto to, in the least-squares sense.

hat

The skew-symmetric matrix of a vector: hat(v) w = v × w.

hat_se3

The embedding of se(3) into 4×4 matrices: ξ^ = [[φ^, ρ], [0, 0]].

inverse_left_jacobian_so3

Inverse left Jacobian of SO(3).

left_jacobian_so3

Left Jacobian of SO(3).

vee

The inverse of hat: recovers the vector from a skew-symmetric matrix.

vee_se3

The inverse of hat_se3.