On convexification of polygons by pops

Abstract

Given a polygon P in the plane, a pop operation is the reflection of a vertex with respect to the line through its adjacent vertices. We define a family of alternating polygons, and show that any polygon from this family cannot be convexified by pop operations. This family contains simple, as well as non-simple (i.e., self-intersecting) polygons, as desired. We thereby answer in the negative an open problem posed by Demaine and O'Rourke [9, Open Problem 5.3].

Convexification of polygons by length preserving transformations

“…Until recently, it has remained an open problem whether there exist polygons that cannot be convexified by pops [7,Open Problem 5.3]. We have shown [8] that such polygons do indeed exist, from both classes, simple or self-intersecting, thereby answering the above open problem in its full generality.…”

“…There is an extensive bibliography pertaining to these subjects [7]. See also [6,8] and the references therein. More results on edge-length preserving transformations and chord stretching can be found in [4,12,13,14].…”

The four bars problem

“…with H 12 (θ 1 , θ 2 ) : (−π, π] → R. The discrete time map (7) describes the change of angles induced by popping bars 1-2. For bars 2-3, it follows on similar lines that…”

“…The two-dimensional maps (7) and ( 8) describe the behavior of all four-bar linkages characterized by the lengths l 1 , l 2 , and l 3 . In order to consider the behavior of a unique four-bar linkage, one has to treat the length L of the fixed bar (not used to derive ( 7)-( 8)) as an additional constraint.…”

“…The question of whether polygons can be convexified by a (finite) series of pops, under various intersection restrictions and definitional variants is studied by Aloupis et al [1]. Dumitrescu and Hilscher show that not every polygon can be convexified with pops [7]. However, their counterexample is highly degenerate (all vertices of the polygon lie on two orthogonal lines).…”