SE3se3

SE(3) and se(3) refer to the special Euclidean group in three dimensions and its Lie algebra, the foundational mathematical framework for 3D rigid body motions and their infinitesimal generators. SE(3) consists of all rigid transformations that combine rotation and translation, while se(3) captures the infinitesimal motions (twists) that generate these transformations.

An element of SE(3) is typically represented as a 4x4 homogeneous matrix g = [R t; 0 1], where R is a 3x3 rotation matrix in SO(3) and t is a 3-vector for translation in R^3. The group operation is matrix multiplication; the identity element is [I 0; 0 1], and the inverse is [R^T −R^T t; 0 1]. SE(3) acts on 3D points p in R^3 via p' = R p + t, describing the pose of a rigid body.

The Lie algebra se(3) comprises 4x4 matrices of the form ξ_hat = [ω^ v; 0 0], where ω ∈ R^3 represents angular velocity and v ∈ R^3 represents linear velocity per unit rotation. Here ω^ denotes the 3×3 skew-symmetric cross-product matrix corresponding to ω. The six-dimensional twist ξ = (ω, v) generates infinitesimal rigid motions; the Lie bracket encodes the screw-theoretic interactions between twists.

The exponential map exp: se(3) → SE(3) relates twists to finite motions. For a twist ξ = (ω, v) with parameter θ, the resulting pose is g = exp(ξ_hat θ), yielding rotation R = exp(ω^ θ) and translation p given by a closed-form expression that reduces to p when ω = 0. The adjoint representation Ad_g describes how twists transform between coordinate frames.

Applications of SE(3) and se(3) span robotics, computer vision, SLAM, and computer animation, where they model pose, motion, and the kinematics of rigid bodies. The term SE3se3 is not standard; it is sometimes used informally to refer to discussions or data structures that encompass both the pose (SE(3)) and its infinitesimal generators (se(3)).