By stereographic projection we can assume that the given Jordan arc lies on a sphere, and that its two ends are at opposite poles, $N$ and $S$. Now project the arc onto a cylinder that touches the sphere at the equator, so that $N$ and $S$ go to infinity on opposite ends of the cylinder.
By stereographic projection we can assume that the given Jordan arc lies on a sphere, and that its two ends are at opposite poles, $N$ and $S$. Now project the arc onto a cylinder that touches the sphere at the equator, so that $N$ and $S$ go to infinity on opposite ends of the cylinder.