A Novel Representation of 3D Rotations: The Gibbs-Rodrigues Representation

Title: A Novel Representation of 3D Rotations: The Gibbs-Rodrigues Representation

Abstract: This research explores a little-known representation of rotations in a three-dimensional space called the Gibbs-Rodrigues representation. The study demonstrates a set of algorithms for handling this representation and compares it to other methods like Euler angles and quaternions. The main findings suggest that the Gibbs-Rodrigues representation is more efficient and computationally friendly than Euler angles, has affinities with Hassenpflug's Argyris angles, and is closely related to the quaternion representation. While the quaternion representation avoids the discontinuities inherent in any three-dimensional representation, the paper presents efficient algorithms for dealing with these issues in the Gibbs-Rodrigues representation.

Main Research Question: Can the Gibbs-Rodrigues representation be an efficient and practical method for representing 3D rotations?

Methodology: The study investigates the Gibbs-Rodrigues representation by analyzing its properties and comparing it to other methods. It presents a set of algorithms for transforming between the representation and a rotation matrix, and for determining rotations based on geometrical constraints.

Results: The research finds that the Gibbs-Rodrigues representation is a computationally efficient way of representing a rotation with three values. Transformation of the coordinate system causes the three values taken together to transform like the components of a vector, which is parallel to the rotation axis. Non-transcendental algorithms exist for transforming back and forth between the matrix and vector forms, and for determining the vector representing a rotation defined in terms of commonly-encountered geometrical constraints.

Implications: The Gibbs-Rodrigues representation offers a practical and efficient alternative to existing methods like Euler angles and quaternions for representing 3D rotations. It has applications in various fields such as computer graphics, robotics, and biology, where handling rotations in three-dimensional spaces is common. The algorithms presented in the paper can be used to implement the representation in computer programs, providing a more efficient way of handling rotations.

Link to Article: https://arxiv.org/abs/0104016v4 Authors: arXiv ID: 0104016v4