Representations of 3D Rotations: Mathematical Foundations and Comparative Analysis

This research paper comprehensively investigates diverse representations of the special orthogonal group SO(3), such as Euler angles, axis-angle vectors, quaternions, rotation matrices, exponential maps, and emerging continuous and probabilistic methods. The study evaluates their mathematical formulations, continuity, susceptibility to gimbal lock, computational efficiency, storage requirements, interpolation properties, and composition operations, integrating algebraic insights with practical applications in animation, pose estimation, inertial navigation, 3D shape registration, and neural networks. Empirical evidence highlights quaternions' dominance due to their compactness and computational efficiency, while alternatives like 6D continuous representations and matrix Fisher distributions provide enhanced continuity and uncertainty modeling.