Shapes of surfaces that contain a great and a small circle through each point

TL;DR

The paper classifies real surfaces in the 3-sphere $S^3$ that contain a great circle and a small circle through every general point, introducing a projective elliptic-geometry framework and a sectional delta invariant to control singularities. It proves that degree-eight great celestial surfaces fall into three shapes (I–III), realizable as $A_0\star B_1$, $A_0\star B_2$, or $A_0\star B_3$, and provides a topological classification into five normal forms, including configurations of two tori or a torus with a circle. The approach combines a Möbius-quadratic model, divisor-class analysis on singular surfaces, and central projection to recover singular loci and incidences between circles, culminating in a complete shape classification with explicit parametrizations. These results advance the understanding of celestial surfaces in $S^3$, with potential applications in geometric modeling and architectural design where circular patches and interconnections are important.

Abstract

We classify the topological types of surfaces in the 3-dimensional unit sphere that contain both a great and a small circle through each point. In particular, these surfaces are homeomorphic to one of five normal forms and are either the pointwise product of circles in the unit quaternions or contain five concurrent circles. We classify the real singular loci of such surfaces and characterize how circles in the surface meet the self-intersection locus.

Figures (8)

• Figure 1: Stereographic projections of surfaces in $S^3$ that contain a great (red) and a small (blue) circle through each point.
• Figure 2: Two stereographic projections of a surface in $S^3$ that has Shape \ref{['I']}.
• Figure 3: Each line segment represents a complex line in $\tau({\mathcal{X}})$. Two line segments meet at a disk if and only if their corresponding complex lines intersect.
• Figure 4: Each line segment represents a complex double line and the green loop represents a great double circle in ${\mathcal{X}}$. The line segments and/or loop meet at a disk if and only if their corresponding components in $\operatorname{Sing}\tau(X)$ intersect.
• Figure 5: See the proof of \ref{['prp:star']}. Each curve segments correspond to one of the complex curves $A,B,C,R,\overline{R}\subset {\mathcal{X}}$ or $F_\beta\subset{\mathcal{Z}}$, where $\beta\in B_{\mathbb{R}}$. Two curve segments meet at a disk if and only if the corresponding complex curves intersect. The incidence point $\varepsilon\in{\mathbb{S}}^3_{\mathbb{R}}$ corresponds via $\mathbf{R}$ to the identity quaternion $\mathbf{1}\in S^3$.

Theorems & Definitions (64)

• Corollary I
• Corollary II
• Definition 1
• Corollary III
• Theorem I
• Example 2
• Remark 3
• Conjecture 1
• Proposition 4
• proof