An Alternative Approach to Elliptical Motion

TL;DR

This paper introduces a novel framework for generating elliptical rotations by defining elliptical inner products, matrices, and quaternions, extending classical rotation methods to elliptical geometries with proofs and numerical examples.

Contribution

It develops new elliptical rotation matrices using elliptic inner products, skew symmetric matrices, and elliptic quaternions, providing a comprehensive alternative to traditional methods.

Findings

Elliptic rotation matrices are constructed via elliptic Rodrigues, Cayley, and Householder methods.

Elliptic quaternions are defined and used to generate elliptical rotation matrices.

Numerical examples validate the proposed elliptical rotation methods.

Abstract

Elliptical rotation is the motion of a point on an ellipse through some angle about a vector. The purpose of this paper is to examine the generation of elliptical rotations and to interpret the motion of a point on an elipsoid using elliptic inner product and elliptic vector product. To generate an elliptical rotation matrix, first we define an elliptical ortogonal matrix and an elliptical skew symmetric matrix using the associated inner product. Then we use elliptic versions of the famous Rodrigues, Cayley, and Householder methods to construct an elliptical rotation matrix. Finally, we define elliptic quaternions and generate an elliptical rotation matrix using those quaternions. Each method is proven and is provided with several numerical examples.