Surfaces containing two circles through each point and Pythagorean 6-tuples

TL;DR

This paper investigates special analytic surfaces in 3D space that contain two circles through each point, linking geometric properties to algebraic Pythagorean 6-tuples via quaternionic parametrization.

Contribution

It introduces a novel algebraic approach to classify such surfaces by reducing the problem to finding Pythagorean 6-tuples of polynomials, extending classical geometric methods.

Findings

Reduction of surface classification to algebraic Pythagorean 6-tuples

Use of quaternionic rational parametrization for surface analysis

Connection to Darboux's classical problems

Abstract

We study analytic surfaces in 3-dimensional Euclidean space containing two circular arcs through each point. The problem of finding such surfaces traces back to the works of Darboux from XIXth century. We reduce finding all such surfaces to the algebraic problem of finding all Pythagorean 6-tuples of polynomials. The reduction is based on the Schicho parametrization of surfaces containing two conics through each point and a new approach using quaternionic rational parametrization.