Solution of a quadratic quaternion equation with mixed coefficients

TL;DR

This paper thoroughly analyzes the solution space of a specific quadratic quaternion equation with mixed coefficients, revealing geometric structures like circles and spheres in quaternion space.

Contribution

It provides a comprehensive classification of solutions for a special quadratic quaternion equation, including geometric interpretations and solution conditions.

Findings

Solutions can be two, one, or no solutions for generic coefficients.

Special cases yield solutions forming a circle or a 3-sphere in quaternion space.

The analysis offers geometric insights into quaternion equations.

Abstract

A comprehensive analysis of the morphology of the solution space for a special type of quadratic quaternion equation is presented. This equation, which arises in a surface construction problem, incorporates linear terms in a quaternion variable and its conjugate with right and left quaternion coefficients, while the quadratic term has a quaternion coefficient placed between the variable and its conjugate. It is proved that, for generic coefficients, the equation has two, one, or no solutions, but in certain special instances the solution set may comprise a circle or a 3-sphere in the quaternion space $\mathbb{H}$. The analysis yields solutions for each case, and intuitive interpretations of them in terms of the four-dimensional geometry of the quaternion space $\mathbb{H}$.