Closing curves by rearranging arcs

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.