The Gage-Grayson-Hamilton curve-shortening flows along the normal to the curve:
(Image from Andrejs Treibergs presentation PDF.)
The result is remarkable avoidance of self-intersection, regardless of how convoluted is the original curve, and ultimate convergence to a "round point." See, e.g., this YouTube Video.
My question is:
Q. Has this flow been studied along vectors at some fixed angle to the normal? E.g., at $90^\circ$ counterclockwise of the normal, tangent to the curve?
Presumably, for any angle $\alpha > 0$ counterclockwise of the normal, the curve might self-intersect during its evolution. But perhaps it still converges to a limit shape?