Vertex Pops and Popturns

. Pops and popturns are polygonreconfiguration moves similar to “Erdos pocket flips”and “flipturns” [5, 3] in that they preserve the lengthsof the polygon edges.Our goal in this paper is to study which polygons canbeconvexifiedbyaseriesofpopsorpopturns, undervar-ious intersection restrictions and definitional variants.We distinguish between three types of polygons. Asimple polygon is non-self-intersecting, in that edges in-tersect only at common endpoints. A polygon is weaklysimple if its boundary does not “properly cross” itself.Finally, a general polygon may be self-intersecting withproper crossings. Pops and popturns can easily intro-duce weak or proper crossings, so the latter two classesare often more natural to study.We also focus on two subclasses of polygons. In anorthogonal polygon, adjacent edges meet at right angles.In an equilateral or unit polygon, all edge lengths areequal, say, to 1. In unit polygons, pops and popturnsbecome identical operations.We will see that a vertex pop can create a hairpin ver-tex (or a pin): a vertex v