Central figure-8 cross-cuts make surfaces cylindrical

TL;DR

This paper proves that certain smooth surfaces in three-dimensional space, which intersect planes along symmetric figure-8 curves and have centrally symmetric compact plane intersections, must be cylindrical over a central figure-8, revealing new geometric properties.

Contribution

It establishes a characterization of surfaces based on their planar intersections and symmetry properties, connecting figure-8 curves and cylindrical surfaces in Euclidean space.

Findings

Surfaces intersecting planes in central figure-8s are cylindrical over such curves.

Centrally symmetric loops with even rotation number are constrained in their symmetry.

New properties of centrally symmetric loops in the plane are demonstrated.

Abstract

We prove: If a complete connected smooth surface M in euclidean 3-space has general position, intersects some plane along a clean figure-8 (a loop with total curvature zero) and all compact intersections with planes have central symmetry, then M is a (geometric) cylinder over some central figure-8. On the way, we establish interesting facts about centrally symmetric loops in the plane; for instance, a clean loop with even rotation number 2k can never be central unless it passes through its center exactly twice and k=0.