3D Rotation Representations: Mathematical Foundations and Comparative Analysis

This paper systematically reviews multiple rotation representations of SO(3), including Euler angles, axis-angle vectors, quaternions, rotation matrices, exponential maps, and emerging continuous and probabilistic methods, and compares them across dimensions such as continuity, susceptibility to gimbal lock, computational efficiency, storage requirements, interpolation, and composition operations. Empirical results show that quaternions are advantageous due to their compactness and efficiency, while 6D continuous representations and matrix Fisher distributions offer greater advantages in continuity and uncertainty modeling.