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

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.

minor stylistic changes; 15 pages, 1 figure

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