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:

. 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?Q

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?

movement in the direction of the tangent only affect the parametrization not the embedding, so projecting your speed vector to its normal part should not change the shape of the evolution. If I find the time I'll try to make it rigorous and post it as an answer. $\endgroup$ – Thomas Richard Nov 13 '16 at 12:39