Convexifying Star-Shaped Polygons

Abstract : We show that all star-shaped simple polygons can be transformed, in O(n) steps each of which takes O(n) time to compute in an extended real RAM model, into convex polygons by continuous motions during which the edges remain rigid and the polygon remains star-shaped and simple. For each step, the trajectories of the vertices are given explicitly. In particular, the vertices move along straight lines, circular-arcs, or on the curve traced by an intersection point of two circles whose centers move along straight lines.