It turns out there is an awesome solution from the field of Topology that involves covering the curve with a self-intersecting Möbius strip.
Want to see how this looks in 3D ?
Feel free to play around:
I was greatly inspired by 3Blue1Brown to create this model. Here is the video where he explains why this solution works: https://www.youtube.com/watch?v=AmgkSdhK4K8