Surfaces containing two circles through each point and decomposition of quaternionic matrices

TL;DR

This paper classifies special surfaces in three-dimensional space that contain two circles through every point, utilizing a novel quaternionic matrix decomposition method to achieve this characterization.

Contribution

It introduces a new quaternionic matrix decomposition technique and applies it to classify surfaces with two circles through each point.

Findings

Classified all such surfaces in R^3.

Developed a new quaternionic matrix decomposition method.

Connected geometric surface properties with algebraic matrix techniques.

Abstract

We find all analytic surfaces in space R^3 such that through each point of the surface one can draw two circular arcs fully contained in the surface. The proof uses a new decomposition technique for quaternionic matrices.