Translational and great Darboux cyclides

TL;DR

This paper classifies translational and great Darboux cyclides, revealing their geometric properties and the number of circles through a general point in various contexts, using combinatorial methods on low degree curves.

Contribution

It provides a comprehensive classification of translational and great Darboux cyclides, detailing the possible configurations of circles through a general point.

Findings

Surfaces as sums of circles are either coplanar or have at most 2 circles through a point.

Surfaces as products of circles in quaternions have 2 to 5 circles through a point.

Great sphere surfaces contain 4 to infinitely many circles through a point.

Abstract

A surface that is the pointwise sum of circles in Euclidean space is either coplanar or contains no more than 2 circles through a general point. A surface that is the pointwise product of circles in the unit-quaternions contains either 2, 3, 4, or 5 circles through a general point. A surface in a unit-sphere of any dimension that contains 2 great circles through a general point contains either 4, 5, 6, or infinitely many circles through a general point. These are some corollaries from our classification of translational and great Darboux cyclides. We use the combinatorics associated to the set of low degree curves on such surfaces modulo numerical equivalence.