arc-overhang
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Tesselation Overhang Pattern
Yet another suggestion of a method to consider for generating overhang path.
This is one thing that I came up with when wondering whether it was possible to avoid "single point starts." The idea is that we imagine that there's a triangular tesselation on the overhang section, and then fill it tile-by-tile going breadth first. If 2 or 3 of the neighbors of an empty tile fill on the same step then the shape isn't quite so nice. (Those correspond to the red and blue sections in the image.)
This kind of a approach should prevent the fractal effect that we see as a result of the current greedy approach.