Investigate these modules to see if edge rendering can be done the same way Line2D renders.

[remorse107]

Line2D node code:
https://github.com/godotengine/godot/blob/master/scene/2d/line_2d.cpp

LineBuilder code:
https://github.com/godotengine/godot/blob/master/scene/2d/line_builder.cpp <-- This is where the tessellation part happens.

[stevenctl]

I don't know about porting too much of this. The general approach is ok, but the corner detection breaks down when we have vertices really close together.