Is there a theoretical (or practical) definition of $n$-gon, for $n < 0$?

Question

Background

This is purely a "sate my curiosity" type question.

I was thinking of building a piece of software for calculating missing properties of 2D geometric shapes given certain other properties, and I got to thinking of how to failsafe it in case a user wants to calculate the area of a $2$-gon, $1$-gon, $0$-gon, 'aslkfn'-gon or maybe even $-4$-gon.

Question

Are there any definitions for $n$-gons where $n < 0$?

Valid assumptions

Let's, for the sake of simplicity (if possible) say that $n \in \mathbb Z$, although I might come back later and ask what a $\pi$-gon is.

Answer 1

For regular $n$-gons with side-length $1$, the area is given as $$\frac{1}{4}n \cot \frac{\pi}{n}$$

Here are some values of the formula for negative values of $n$:

\begin{array}{c|c} n & \cot\frac{\pi}{n} \\ \hline -1 & \text{complex } \infty \\ -2 & 0\\ -3 & -\frac{1}{\sqrt{3}} \\ n \leq 3 & <0 \end{array}

How you want to interpret that is up to you, but the function is there.

This kind of thing (where you extend something beyond what's intuitive) is done many places within mathematics. Take for instance the $\Gamma$-function (see here), which is an extension of the factorial function. However, I don't know if it is useful in this particular case (with the $n$-gons), but why not try?

Answer 2

You may define a digon on $S^{2}$. By the ways, there're $\{ \frac{n}{k} \}$ star polygons. In particular $\{ \frac{n}{1} \}$ or $\{ \frac{n}{n-1} \}$ is ordinary polygon.

Answer 3

For regular n-gons inscribed in a given oriented circle, a regular $(-n)$-gon can be defined to be regular n-gon with the opposite orientation.

This is not a definition I have ever seen in a publication, but it is consistent with most of the standard conventions.

Extending the idea a bit, one can take an orientation of the plane and consider the data of an n-gon, not necessarily regular, to always include a cyclic ordering of the vertices, and $n \to (-n)$ being orientation reversal. Whether a given convex n-gon has positive or negative $n$ can be interpreted as the question of whether its area, in the given cyclic ordering, is positive relative to the orientation of the plane.