SE2

Se2, formally SE(2), is the Special Euclidean group in two dimensions. It describes all rigid motions of the plane, i.e., every combination of a plane rotation and a translation. An element can be written as a pair (R, t) with R ∈ SO(2) a rotation and t ∈ R^2 a translation. In homogeneous coordinates, SE(2) is represented by 3×3 matrices of the form

[ R t ]

[ 0 1 ]

where R = [ [cos θ, −sin θ], [sin θ, cos θ] ] and t = [x, y]^T, with θ, x, y ∈ R. The group composition is the usual combination of motions: (R1, t1) · (R2, t2) = (R1 R2, R1 t2 + t1). SE(2) is three-dimensional, non-abelian, and isomorphic to the semi-direct product R^2 ⋊ SO(2).

Its Lie algebra, se(2), consists of 3×3 matrices of the form

[ 0 −ω v_x ]

[ ω 0 v_y ]

[ 0 0 0 ]

where ω, v_x, v_y ∈ R. The corresponding twist coordinates are (v_x, v_y, ω). The exponential map exp: se(2) → SE(2) yields a rigid motion from a twist. If ω ≠ 0 and θ = ω t, the resulting transformation has rotation R = Rot(θ) and translation t computed by a closed-form formula, for example

t = [ (v_x sin θ + v_y (1 − cos θ)) / ω, (−v_x (1 − cos θ) + v_y sin θ) / ω ]^T.

If ω = 0, the transform reduces to a pure translation t = t_vec with R = I. SE(2) underpins 2D robotics, computer vision, and animation, including 2D pose estimation, planar robot kinematics, and SLAM.