This paper explores the use of continuous representations for rotations in neural networks, addressing the limitations of discontinuous representations such as quaternions and Euler angles. The authors define a continuous representation in the context of neural networks, relating it to topological concepts like homeomorphism and embedding. They investigate the continuity of 2D, 3D, and n-dimensional rotations, finding that all representations are discontinuous in real Euclidean spaces of four or fewer dimensions for 3D rotations. However, they propose continuous representations in 5D and 6D for 3D rotations, which are more suitable for learning. These continuous representations are also applicable to other groups like the orthogonal group and similarity transforms. Empirical results show that the proposed continuous representations outperform discontinuous ones in various tasks, including rotation estimation, pose estimation, and inverse kinematics. The paper concludes by highlighting the advantages of continuous representations in neural networks for rotation tasks.This paper explores the use of continuous representations for rotations in neural networks, addressing the limitations of discontinuous representations such as quaternions and Euler angles. The authors define a continuous representation in the context of neural networks, relating it to topological concepts like homeomorphism and embedding. They investigate the continuity of 2D, 3D, and n-dimensional rotations, finding that all representations are discontinuous in real Euclidean spaces of four or fewer dimensions for 3D rotations. However, they propose continuous representations in 5D and 6D for 3D rotations, which are more suitable for learning. These continuous representations are also applicable to other groups like the orthogonal group and similarity transforms. Empirical results show that the proposed continuous representations outperform discontinuous ones in various tasks, including rotation estimation, pose estimation, and inverse kinematics. The paper concludes by highlighting the advantages of continuous representations in neural networks for rotation tasks.