Vertex Pops and Popturns

This paper considers transformations of a planar polygon P according to two types of operations. A vertex pop (or a pop) reflects a vertex vi, i ∈ {1, . . . , n}, across the line through the two adjacent vertices vi−1 and vi+1 (where index arithmetic is modulo n). A popturn rotates vi in the plane by 180◦ about the midpoint of the line segment vi−1vi+1. Pops and popturns are polygon reconfiguration moves similar to “Erdős pocket flips” and “flipturns” [5, 3] in that they preserve the lengths of the polygon edges. Our goal in this paper is to study which polygons can be convexified by a series of pops or popturns, under various intersection restrictions and definitional variants. We distinguish between three types of polygons. A simple polygon is non-self-intersecting, in that edges intersect only at common endpoints. A polygon is weakly simple if its boundary does not “properly cross” itself. Finally, a general polygon may be self-intersecting with proper crossings. Pops and popturns can easily introduce weak or proper crossings, so the latter two classes are often more natural to study. We also focus on two subclasses of polygons. In an orthogonal polygon, adjacent edges meet at right angles. In an equilateral or unit polygon, all edge lengths are equal, say, to 1. In unit polygons, pops and popturns become identical operations. We will see that a vertex pop can create a hairpin vertex (or a pin): a vertex vi whose incident edges overlap collinearly. If also vi−1=vi+1 (which arises naturally in unit polygons), then the reflection line for a pop of vi is not determined. Whether to allow a pop of such a pin, and if so, how to define it, leads to many interesting variations, detailed in Section 3 below.