  We describe new methods for the construction of spiral toolpaths for
high-speed machining. In the simplest case, our method takes a polygon as input
and a number $\delta>0$ 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 $\delta$. 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. Finally, we
show how to make a spiral in a polygon with multiple islands by connecting the
islands into one island.
