Efficient Vector Representation for 3D Rotations

Title: Efficient Vector Representation for 3D Rotations

Abstract: This research proposes a novel way to represent rotations in three-dimensional space. Unlike the commonly used Euler angles, which require the evaluation of transcendental functions and have ambiguities, this method uses a vector representation that transforms like a vector under changes in coordinates. This representation is more efficient, easier to handle, and has affinities with other well-known rotation representations. The paper presents algorithms for generating and transforming the vector representation, and for determining the rotation that maps a given pair of vectors to another pair of the same length and subtended angle.

Main Research Question: How can we efficiently represent rotations in three-dimensional space using a vector representation that avoids the need for transcendental functions and has fewer ambiguities?

Methodology: The study uses a vector representation for rotations, where the vector is parallel to the axis of rotation and its components transform covariantly on changes of coordinates. This representation is more efficient than Euler angles and has affinities with Hassenpflug's Argyris angles and the quaternion representation. The paper demonstrates algorithms for generating the rotation matrix from the vector representation and vice versa, and for determining the rotation that maps a given vector to another of the same length.

Results: The research shows that the vector representation is more efficient than Euler angles and has fewer ambiguities. The algorithms presented for generating and transforming the vector representation are computationally efficient and avoid the need for transcendental functions.

Implications: The proposed vector representation for 3D rotations offers several advantages over the Euler angle representation. It is more efficient, easier to handle, and has fewer ambiguities. This could lead to improved performance and reliability in applications that involve rotations in three-dimensional space, such as computer graphics, robotics, and molecular biology.

Link to Article: https://arxiv.org/abs/0104016v2 Authors: arXiv ID: 0104016v2