Math & Kinematics

SO(3)

SO(3) is the special orthogonal group in three dimensions: the set of all 3x3 rotation matrices, i.e., orthogonal matrices with determinant one, representing every possible orientation of a rigid body. Common parameterizations include unit quaternions, axis-angle vectors, and Euler angles, each with trade-offs around singularities and double cover. Its Lie algebra so(3) consists of skew-symmetric matrices corresponding to angular velocities, connected to the group by the exponential map.

Why it matters for physical AI

Orientation representation choices ripple through robot learning — regression targets, interpolation, and equivariant architectures all hinge on handling SO(3)'s topology correctly rather than treating rotations as flat vectors.