# Space of simple polygons on $n$-vertices as a set of points in $\mathbb{R}^{2n}$

A simple polygon in $\mathbb{R}^2$ with $n$ vertices can be mapped to elements in $\mathbb{R}^{2n}$ by the list of the coordinates of its vertices. I expect there might be something interesting to study the geometry in this space.

For example, we can qualify how $\varepsilon$-simple polygon is $\varepsilon$ away from a simple polygon in the $\mathbb{R}^{2n}$ space. A geodesic between two points in the space implies some "efficient" continuous transformation that maintains simplicity.

Is there any study on this space?

The field was defined by this seminal paper:

Michael Kapovich and John Millson "On the moduli space of polygons in the Euclidean plane." J. Differential Geom. Volume 42, Number 1 (1995), 133-164.

The topic is now covered to various depths in textbooks, e.g., in Discrete and Computational Geometry:

(Responding to the OP's emphasis on simple (non-self-crossing) polygons:) There has been intense study of the configuration space of simple polygons with the same edge lengths in the same sequence. Thus the edges can be viewed as rigid links, joined at vertices by universal joints. The major result here, by Connolly, Demaine, and Rote, is that this space is connected: one can continuously "morph" between any two instances of a polygon, maintaining simplicity throughout. The proof shows that any polygon can be convexified, and that any two convex instances of the same polygon can be connected. The "doubled tree" example below was at one time thought a possible counterexample:

Animation due to Erik Demaine

Robert Connelly, Erik D. Demaine, and Günter Rote, “Straightening Polygonal Arcs and Convexifying Polygonal Cycles”, Discrete & Computational Geometry, volume 30, number 2, September 2003, pages 205–239. (Author link)

More information on this topic may be found in the book Geometric Folding Algorithms: Linkages, Origami, Polyhedra.

• I'm more interested in simple polygons. I did some googling with this pointer, and it seems there is nothing on moduli space of simple polygons. – Chao Xu Nov 9 '13 at 4:40
• @ChaoXu: There has been work, at least in a certain direction. I added a section above. – Joseph O'Rourke Nov 9 '13 at 14:41