We describe new methods for the construction of spiral tool-paths for high-speed machining. In the simplest case, our method takes a polygon as input and a number δ>0 and returns a spiral starting at a central point in the polygon, going around towards the boundary while morphing to the shape of the polygon. The spiral consists of linear segments and circular arcs, it is G~1 continuous, it has no self-intersections, and the distance from each point on the spiral to each of the neighboring revolutions is at most δ. Our method has the advantage over previously described methods that it is easily adjustable to the case where there is an island in the polygon to be avoided by the spiral. In that case, the spiral starts at the island and morphs the island to the outer boundary of the polygon. It is shown how to apply that method to make significantly shorter spirals in polygons with no islands than what is obtained by conventional spiral toolpaths. Finally, we show how to make a spiral in a polygon with multiple islands by connecting the islands into one island.
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