Translational and great Darboux cyclides

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.

Figures (11)

• Figure 1: Examples of Darboux cyclides.
• Figure 2: $2$-circled surfaces of Möbius degree $8$ (see \ref{['exm:cert']}).
• Figure 3: Incidences between complex lines and isolated singularities on a Darboux cyclide $X$ (see \ref{['exm:Y']}). Each complex line is represented as a line segment and labeled with its corresponding class in $E(X)$. A real or non-real isolated singularity is represented as a disc with a solid and dashed border, respectively. Each singularity is labeled with the sum of classes in the corresponding component in $B(X)$.
• Figure 4: Incidences between complex lines and isolated singularities (see the caption of \ref{['fig:Y']}).
• Figure 5: Incidences between 7 of the 16 lines in a Blum cyclide.

Theorems & Definitions (79)

• Theorem 1
• Remark 1
• Corollary 1
• Corollary 2
• Corollary 3
• Corollary 4
• Conjecture 1
• Remark 2
• Remark 3
• Proposition 4