Is there a hilbert curve equivalent for circles?

Question

Is there a space-filling curve that has the same properties of a hilbert curve (two points close in 1D are close in 2D) but grows in a circular shape instead of a rectangular one?

Answer 1

I am not aware of a such curve construction, but the next best thing is probably using a suitable mapping between square and circle disc and then transform the square based hilbert curve onto a circle.

Maybe something like this: Conformal mapping circle onto square (and back)