3 research outputs found
The Four Bars Problem
A four-bar linkage is a mechanism consisting of four rigid bars which are
joined by their endpoints in a polygonal chain and which can rotate freely at
the joints (or vertices). We assume that the linkage lies in the 2-dimensional
plane so that one of the bars is held horizontally fixed. In this paper we
consider the problem of reconfiguring a four-bar linkage using an operation
called a \emph{pop}. Given a polygonal cycle, a pop reflects a vertex across
the line defined by its two adjacent vertices along the polygonal chain. Our
main result shows that for certain conditions on the lengths of the bars of the
four-bar linkage, the neighborhood of any configuration that can be reached by
smooth motion can also be reached by pops. The proof relies on the fact that
pops are described by a map on the circle with an irrational number of
rotation.Comment: 18 page
On convexification of polygons by pops
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]