# Parameterizing equilateral polygons

I'm not exactly sure how to describe what I want, so if I butcher terms, please forgive me :)

I want to "parameterize" the space of simple irregular equilateral polygons with n sides, or at least a large-ish subset of them. That is, the user provides the number of edges in the polygon ($n$) and a set of parameters $[x_1, x_2, ... x_m]$ that satisfy some equalities and inequalities (so $Ax \ge b, Cx = d$ for $A \in \mathbf{R}^{p \times m}, b \in \mathbf{R}^p, C \in \mathbf{R}^{q \times m}, d \in \mathbf{R}^q$). For each valid configuration vector $x$ I want to be able to construct a unique equilateral $n$-gon. I want the space of polygons I can construct to be as rich as possible.

As an example, if $n$ is 3, then the only valid equilateral 3-gon is the equilateral triangle and the configuration space is basically empty so the whole thing is moot. If $n$ is 4, then valid polygons are convex and shaped like kites, and there's basically only a single degree of freedom and I could use the width, say, as my configuration vector.

For larger values of $n$, though, the obvious possibilities for $x$ start to break down. For instance if you take the sum of the internal angles of the polygon and force them to add to $\pi (n - 2)$ you're not guaranteed that the ends meet up and form a complete circuit. If you try to require that the ends meet up I don't see a way to express that as a linear equality. (Notice that constructing the polygon doesn't have to be a linear operation. Just that the constraints on the configuration vector are linear).

Or if you assume the polygon is a star-shaped polygon, with the values of $x$ being the distances from some "center" point in the kernel to the vertices of the polyogn, I don't see a way to enforce the equilateral-ness of the polygon without adding non-linear constraints.

If you assume the polygon is regular, it gets quite easy of course, but the configuration space isn't very rich. Ideally I'd like to support at least some amount of concave polygons. Limiting it to some large-ish subset of the equilateral star-shaped polygons would be reasonable, for instance. But even being able to parameterize generic convex polygons (maybe as some intersection of half planes) would be interesting.

I'm just not sure what so search for or how to frame the problem mathematically.

• I really like the question... Have you checked for references online though? It seems at least a few things are known: books.google.com/…. And explicitly (but for small n) researchgate.net/publication/… – Theo Mar 31 '15 at 22:09
• The particular paper that is referred to in the book is [ac.els-cdn.com/0166864196000466/… – Theo Mar 31 '15 at 22:13
• I tried searching for things online but I'm apparently not very good at it haha :) I'll look at the links you give. What search terms are you using to search with? – Jay Lemmon Mar 31 '15 at 22:14
• Just try "Configuration space of equilateral polygons" – Theo Mar 31 '15 at 22:15