On the Deflation of Polygons

A balloon being deflated.

Introduction

The deflation of polygons is essentially the inverse to the problem of polygon inflation (also considered here). In 1939, Bela Nagy proved Erdos's conjecture that every simple polygon can be made convex in a finite number of inflations. Then in 1993, Bernd Wegner proposed that every simple polygon can be fully deflated after a finite number of deflations.

We will see that there are some polygons which (unlike the case of inflation) allow an infinite number of deflations.

Definitions

Formally speaking, a deflation P' of a polygon P is a new polygon constructed from P in the following way. Choose vertices p_i and p_j of P such that p_i is not adjacent to p_j (that is, p_i is not equal to p_{j + 1} or p_{j - 1}). Also, the entire subarc of P from p_i to p_j must be completely on one side of the line from p_i to p_j. As well, the line from p_i to p_j must not be a "line of support". That is, it must not be completely to one side of P. This line is known as a cut. Then for all points p_n where i < n < j, reflect p_n around the line between p_i and p_j. The deflation is legal if the resulting polygon P' is simple. If there is no possible legal deflation on a polygon, then the polygon is said to be deflated.

So, as we see, there are cuts that are completely invalid, as shown here:

Obviously, this is would be valid as a line of reflection in an inflation, but not in a deflation. Also, there are cuts that are valid, like

but with deflations that are not legal

Finally, thare are valid cuts

that have legal deflations

This polygon is deflated after only two deflations.

Try out some deflations with this random polygon. To deflate the polygon across a given cut, left-click the points that you want to reflect across. If the resulting polygon would not be simple, nothing will happen. Otherwise, you will see the deflation. To get a new random polygon, right-click anywhere on the applet.

The counter example to the conjecture that each polygon has finitely many deflations

Consider a polygon with four vertices labeled A through D in counterclockwise order. Then if the following two conditions hold, the polygon may be deflated an infinite number of times.

1. The sum of the lengths of the edges AB and CD is equal to the sum of the lengths of the edges BC and DA.
2. No two adjacent edges have the same length.

Proof: First, we will see that such a polygon retains the above qualities on each deflation. Then, we will see that any such polygon is simple.

The first statement is true since a deflation does not change the lengths of any of the edges. Now assume that the second statement is false. If this was the case, then a pair of opposite edges intersect. However, this means that either AD + BC > AB + CD or AD + BC < AB + CD by the triangle inequality.

Why? Consider the figure below where AB crosses CD at point x. Then the triangle inequality says that Ax + xC > AC and also that Dx + xB > DB. Adding those two inequalities, we get Ax + xC + Dx + xB > AC + DB. However Ax + xB = AB and xC + Dx = CD, so we get that AB + CD > AC + DB. Also, these inequalities must be strict in order for condition 2 to hold. This contradicts condition 1.

A non simple quadrilateral.

Therefore, there are polygons with infinitely many deflations.

Applet

Click on the applet below and watch the polygon deflate without end. Eventually, it will degenerate into a line. At that point, clicking it will not appear to have much effect. You can reset the polygon by right clicking on the applet.

The code is made up of three files: MyPolygon.java, the interesting one, Point.java, a simple point class which implements a heap sort to make semi-random polygons with, and DeflationApplet.java, the part that implements the user interface such as it is. The class to compare the points in the heapsort is found in PointComparer.java

This applet shows a polygon with sides proportional to 2, 3, 2, and 1. Thus, this polygon is infinitely deflatable. Each deflation is just a single reflection. The reflection is found by computing the point q on the line of reflection that is nearest to the point that is being reflected. Then the difference between q and and the point being reflected is added to q and assigned to the point being reflected.

References

Thomas Fevens, Antonio Hernandez, Antonio Mesa, Patrick Morin, Michael Soss and Godfried T. Toussaint, "Simple polygons with an infinite sequence of deflations" Contributions to Algebra and Geometry, Vol. 42, No. 2, 2001, pp. 307-311.

Bernd Wegner, "Partial Inflation of Closed Polygons in the Plane", Contributions to Algebra and Geometry, Vol. 34, No. 2, 1993, pp. 77-85.