In order to simplify the exposition of the
permutation problem, we will only look at signed simple polygons.

Given a signed simple planar polygon with a sequence of edges e = {e1,
e2, ...., en} we examine the effects of several operations that permute
this sequence, resulting in the formation of a new polygon. The purpose
is to study the complexity of performing certain actions such as **convexifying**
a given polygon or obtaining the mirror image by using these permuting
operations.

Some form of restrictions must be imposed upon those operations so that
we do not deteriorate the goals of this research. One of which is the
conservation of simplicity. In other words applying a permutation on
a simple polygon must produce another simple polygon, therefore some
polygons do not admit certain type of operations. Here an example:

*Polygon that do not admit any edge-swaps. *

**There's 3 main operations that we can consider:**

**Reversal**:
Inverting the order of a subsequence

* Geometrically equivalent to rotating the subsequence rigidly in
the plane by an angle of 180 so that its endpoints are placed exactly
at each other’s original location.

* A more restricted class of reversals is called Filpturns. We are
going to use Filpturns later in order to prove an upperbound on the
number of reversals required to convexify any simple polygon.

**Transposition:** Interchanging
the edges between two subsequences (no edge sharing allowed).

* Conserves the orientation of a signed polygon

**Edge-Swap:** Special case of a
Transposition, it consist of interchanging two consecutive edges.

* Conserves the orientation of a signed polygon