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The Japanese Theorem for Nonconvex Polygons - The Space of Cyclic Polygons

David Richeson

The Space of Cyclic Polygons

The Japanese theorem tells us that we may associate a single numerical value, \(r_P\), to each cyclic polygon \(P\). In other words, we have found a real-valued function on the space of cyclic polygons. Let us explore this idea a little further.

For simplicity, assume that all of our polygons are inscribed in a circle of fixed radius \(R\) with its center at the origin. We denote the space of cyclic \(n\)-gons \(\mathcal{P}_{R,n}^c = \mathcal{P}_n^c\) (the superscript \(c\) refers to the fact that these are convex polygons, we'll consider nonconvex ones later) and define the function \(f:\mathcal{P}_n^c \rightarrow \mathbb{R}\) to be \(f(P) = r_P\).

Let us be more precise about our definition of \(\mathcal{P}_n^c\). Consider the cyclic polygon determined by the vertices \(p_1, \ldots ,p_n\) labeled counterclockwise, with \(p_1\) fixed at \((R,0)\). For technical reasons we will allow vertices to coincide (e.g., \(p_i = p_{i+1}\)); when this happens we call the associated polygon a degenerate \(n\)-gon because it has fewer than \(n\) sides.

We identify each polygon in \(\mathcal{P}_n^c\) with the vector of central angles \(\theta_1, \ldots, \theta_n \) as in Figure 7. That is, if \(p_k = (R \cos (\alpha_k), R \sin (\alpha_k))\), then \(\theta_k = \alpha_{k+1} - \alpha_k \) for \(k = 1, \ldots, n-1\) and \(\theta_n = 2 \pi - \alpha_n .\) In this way, we may express

\[\mathcal{P}_n^c = \{(\theta_1, \ldots, \theta_n) \in [0, 2 \pi]^n: \theta_1+ \cdots + \theta_n = 2 \pi\}.\]

If we want to exclude the degenerate polygons, then we insist that \(\theta_k > 0\)  for all \(k\).

Figure 7


Thus the space of cyclic \(n\)-gons is an \((n-1)\)-dimensional simplex with the interior of the simplex corresponding to the nondegenerate polygons. For example the space of cyclic triangles is an equilateral triangle and the space of cyclic quadrilaterals is a tetrahedron. A vertex of this simplex corresponds to the degenerate polygon for which \(p_1 = \cdots = p_n\); we can safely ignore the fact that these \(n\) points, \((2 \pi, 0, \ldots, 0), (0, 2 \pi, 0, \ldots, 0), \ldots, (0,\ldots, 0, 2 \pi) ,\) all correspond to the same polygon. Similarly, an edge corresponds to \(n - 1\) vertices equal and one different, and, in general, a \(k\)-dimensional facet corresponds to \(\{ p_1, \ldots, p_n\} \) being \(k + 1\) distinct points on the circumcircle.

David Richeson, "The Japanese Theorem for Nonconvex Polygons - The Space of Cyclic Polygons," Loci (December 2013)