Surfaces containing two circles through each point

TL;DR

This paper classifies all analytic surfaces in three-dimensional space that contain two transversal circles through each point, revealing they are related to specific geometric sets and employing a novel quaternionic polynomial factorization technique.

Contribution

It provides a complete classification of such surfaces, connecting classical geometric problems with modern algebraic methods and quaternionic polynomial factorization.

Findings

Surfaces are images of specific geometric sets under inversions.

Classification includes surfaces related to circles in space and on the sphere.

Introduces a new factorization method for quaternionic polynomials.

Abstract

We find all analytic surfaces in space $\mathbb{R}^3$ such that through each point of the surface one can draw two transversal circular arcs fully contained in the surface. The problem of finding such surfaces traces back to the works of Darboux from XIXth century. We prove that such a surface is an image of a subset of one of the following sets under some composition of inversions: - the set $\{\,p+q:p\in\alpha,q\in\beta\,\}$, where $\alpha,\beta$ are two circles in $\mathbb{R}^3$; - the set $\{\,2\frac{[p \times q]}{|p+q|^2}:p\in\alpha,q\in\beta,p+q\ne 0\,\}$, where $\alpha,\beta$ are two circles in ${S}^2$; - the set $\{\,(x,y,z): Q(x,y,z,x^2+y^2+z^2)=0\,\}$, where $Q\in\mathbb{R}[x,y,z,t]$ has degree $2$ or $1$. The proof uses a new factorization technique for quaternionic polynomials.