The R3 Spin Group

This monograph develops a computational and geometric framework for three-dimensional rotations based on Rotation Vectors — elements of ℝ³ whose direction gives the rotation axis and whose magnitude gives the rotation angle. The central result is a closed-form composition formula that combines two rotations directly in axis-angle form, without requiring conversion to quaternions or matrices. The framework distinguishes sequential composition (for rigid hierarchies and articulated chains) from additive combination (for simultaneous torques and angular velocities), and shows that the sign of a single cross-product term controls the distinction between intrinsic and extrinsic rotation. The presentation connects the Rotation Vector to the Lie algebra 𝔰𝔬(3), the exponential map to Spin(3), and the double-cover structure of the rotation group, while keeping the axis-angle content geometrically visible throughout. Appendices provide the full rotation matrix, matrix recovery formulas, local frame operations, an iterative inverse kinematics method expressed in rotation vector language, and a treatment of the covering structure and winding behavior.