Representations of 3D Rotations: Mathematical Foundations and Comparative Analysis

Rotation representations are foundational in fields such as computer graphics, robotics, and machine learning, where precise and efficient modeling of 3D orientations is critical. This 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, evaluating their mathematical formulations, continuity, susceptibility to gimbal lock, computational efficiency, storage requirements, interpolation properties, and composition operations, while integrating detailed algebraic insights with practical applications in fields like 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. Future research could explore hybrid methods and thorough large-scale evaluations to help build a solid foundation for improving rotation representation techniques.