Surfaces containing two circles through each point

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.

26 pages, 1 figure; this consolidates updates of arXiv:1512.09062 and arXiv:1503.06481, and incorporates a correction to the published version

References in corpus (1)

• Factorizations of Elements in Noncommutative Rings: A Survey

Cited by in corpus (5)

• Factorization of Quaternionic Polynomials of Bi-Degree (n,1)
• Surfaces containing two isotropic circles through each point
• Möbius automorphisms of surfaces with many circles
• Self-intersections of surfaces that contain two circles through each point
• Shapes of surfaces that contain a great and a small circle through each point