Efficient Vector Representation for Three-Dimensional Rotations

Title: Efficient Vector Representation for Three-Dimensional Rotations

Abstract: This research proposes a novel method for representing rotations in three-dimensional space using a set of algorithms based on a three-dimensional vector. The vector is parallel to the axis of rotation and its components transform covariantly with changes in coordinates. This representation has advantages over existing methods like Euler angles, as it is more efficient, does not require transcendental functions, and can be used to generate rotation matrices and vice versa. The paper also discusses the discontinuities in the representation and provides efficient algorithms for mapping vectors to rotations and back.

Main Research Question: How can we efficiently represent rotations in three-dimensional space using a set of algorithms based on a three-dimensional vector?

Methodology: The study uses a three-dimensional vector representation for rotations. The vector is parallel to the axis of rotation and its components transform covariantly on changes of coordinates. The mapping from rotations to vectors is one-to-one, except for computation error, and the rotation matrix can be generated efficiently from the vector without the use of transcendental functions. The paper also discusses the discontinuities in the representation and provides efficient algorithms for computing the set of rotations that map a given vector to another of the same length and the rotation that maps a given pair of vectors to another pair of the same length and subtended angle.

Results: The research demonstrates that the proposed vector representation is more efficient than Euler angles, has affinities with Hassenpflug's Argyris angles, and is closely related to the quaternion representation. The algorithms provided for converting between the vector and rotation matrix representations are efficient and do not require transcendental functions.

Implications: The efficient vector representation for three-dimensional rotations proposed in this research has several implications. It provides a more efficient way of handling rotations in computer representations of three-dimensional space, improves the handling of rigid body movements, orientation of amino and nucleic acid residues in biological polymers, and the attitude of e.g., robot-held tools. Furthermore, the algorithms provided for converting between the vector and rotation matrix representations can be used in various fields that require efficient handling of rotations in three-dimensional space.

Link to Article: https://arxiv.org/abs/0104016v3 Authors: arXiv ID: 0104016v3