Se3Edit
Se3, or SE(3) in standard mathematical notation, is the Lie group of orientation-preserving rigid motions of three-dimensional space. It encodes all possible poses of a rigid body in 3D, combining three degrees of freedom for translation with three for rotation. As the natural configuration space for rigid-body motion in the physical world, SE(3) underpins a wide range of engineering, robotics, computer vision, and graphics applications. It sits at the intersection of geometry and mechanics, and its structure as a six-dimensional manifold makes it a central object in modern applied mathematics.
SE(3) is often described as the connected component of the full Euclidean group E(3) consisting of all isometries that preserve orientation. More concretely, SE(3) can be viewed as the semi-direct product of the rotation group SO(3) and the translation group R^3: SE(3) ≅ SO(3) ⋉ R^3. This means a typical element is a pair (R, t) with R ∈ SO(3) and t ∈ R^3, and composition follows the rule (R1, t1)·(R2, t2) = (R1R2, R1t2 + t1).
Structure and representations
Group structure
The defining property of SE(3) is composition of motions: applying two rigid motions in sequence yields another rigid motion. The orientation of the body is described by a rotation, while its position is described by a translation. The non-commutative, six-dimensional nature of SE(3) reflects the fact that doing a rotation and then a translation generally differs from doing the translation first and then the rotation.
Matrix representation
A standard way to realize SE(3) is through homogeneous transformation matrices of size 4×4: [ [R, t], [0, 1] ] Here R ∈ SO(3) represents rotation and t ∈ R^3 represents translation. Matrix multiplication of these 4×4 blocks corresponds precisely to composing rigid motions. This matrix form is widely used in fields such as robotics and computer graphics, where it provides a convenient framework for chaining multiple motions.
Lie algebra and the exponential map
The tangent space of SE(3) at the identity is the six-dimensional Lie algebra se(3). Elements of se(3) can be represented as 4×4 matrices of the form [ [ω^×, v], [ 0 , 0] ] where ω ∈ R^3 is the angular velocity vector, ω^× is its skew-symmetric cross-product matrix, and v ∈ R^3 is the translational part. The exponential map exp: se(3) → SE(3) takes a twist (ω, v) to a finite rigid-motion, yielding screw motions that combine rotation about an axis with translation along that axis. This logarithmic/exp-log structure is central to many numerical methods in robotics and computer vision.
Parameterizations and numerical considerations
Rotations inside SE(3) can be parameterized in several ways. Euler angles are intuitive but suffer from gimbal lock, making certain sequences numerically unstable. Axis-angle representations and quaternions (for the rotational part) offer more robust alternatives. The choice of parameterization affects optimization, filtering, and estimation tasks in applications like [ [Robot kinematics|robotics] ] and [ [Structure from motion|computer vision] ]. In practice, SE(3) computations often rely on the 4×4 homogeneous form or on compact Lie-algebra methods that preserve the group structure during numerical integration.
Geometry of motions
Chasles’ theorem states that any rigid-body motion in 3D can be represented as a rotation about some axis combined with a translation along a line — a screw motion. This perspective is naturally expressed in the SE(3) framework, linking the geometric intuition of rotations and translations to the algebraic structure of the group and its Lie algebra. The notion of a twist in se(3) captures this idea succinctly and underpins many algorithms for estimating and manipulating poses.
Applications and contexts
Robotics and navigation
SE(3) serves as the standard configuration space for the pose of a robot end-effector, a drone, or a vehicle moving through space. Pose estimation, state estimation, and trajectory planning all rely on SE(3) representations to model how the body moves and how measurements relate to that motion. See also Robot kinematics.
Computer vision and graphics
In computer vision, SE(3) describes the relationship between different camera poses and scene points, enabling 3D reconstruction, camera calibration, and motion tracking. In computer graphics, SE(3) frames are used to place and animate objects consistently in a scene. See also Structure from motion and 3D graphics.
Mathematics and mechanics
From a mathematical perspective, SE(3) is a prototypical example of a Lie group with a nontrivial Lie algebra, illustrating how continuous symmetries combine with translational motion. In rigid-body dynamics, SE(3) provides the natural arena for formulating equations of motion and conservation laws.
Related groups and contrasts
SE(3) is the orientation-preserving part of the full Euclidean group E(3). The full group allows improper rotations (reflections) as well, which leads to richer but more complex structure. In two dimensions, the analogous group is SE(2), which governs planar motions. See also Euclidean group and Special Orthogonal Group.