Closing curves by rearranging arcs
Volume 169 / 2022
Colloquium Mathematicum 169 (2022), 197-208
MSC: Primary 53A04.
DOI: 10.4064/cm8266-6-2021
Published online: 22 February 2022
Abstract
We show how, under surprisingly weak assumptions, one can split a planar curve into three arcs and rearrange them (matching tangent directions) to obtain a closed curve. We also generalize this construction to curves split into $k$ arcs and comment on what can be achieved by rearranging arcs for a curve in higher dimensions. Proofs involve only tools from elementary topology, and the paper is mostly self-contained.
