1A Vector Representation of 3D Rotations

Title: 1A Vector Representation of 3D Rotations

Abstract: This research proposes a new way to represent rotations in three-dimensional space. Instead of using the traditional Euler angles, which involve complex calculations and are not intuitive, the researchers suggest using a three-dimensional vector that is parallel to the axis of rotation. This vector-based representation is covariant, meaning it transforms like a vector under a change of coordinates, and is computationally efficient. The paper provides algorithms for converting between the vector and matrix forms of a rotation and for determining the vector representing a rotation given specific geometrical constraints.

Main Research Question: Can we develop a more efficient and intuitive way to represent rotations in three-dimensional space?

Methodology: The researchers propose using a three-dimensional vector to represent rotations. This vector is parallel to the axis of rotation and its components transform covariantly on a change of coordinates. They provide algorithms for converting between the vector and matrix forms of a rotation and for determining the vector representing a rotation given specific geometrical constraints.

Results: The researchers demonstrate that their vector-based representation is 1:1 apart from computation error and does not require the use of transcendental functions. They also show that efficient algorithms exist for generating the rotation matrix from the vector and vice versa.

Implications: The new vector representation offers several advantages over the traditional Euler angles. It is more intuitive, requires fewer computational resources, and is less prone to errors. This could have significant implications for fields that rely on handling rotations in three-dimensional space, such as computer graphics, robotics, and molecular biology.

Link to Article: https://arxiv.org/abs/0104016v1 Authors: arXiv ID: 0104016v1