Near ideal decompositions of ideal polygons

TL;DR

This work studies moduli spaces of ideal polygons by orthogeodesic decompositions, proving that every ideal $n$-gon has a decomposition into $n-3$ orthogeodesics of length at most $O_n$, with explicit bounds $2 \operatorname{arcsinh}\left(\frac{3}{2} \cot\left(\frac{\pi}{n}\right)\right) \le O_n \le 2 \operatorname{arccosh}\left(\frac{1}{\sin\left(\frac{\pi}{n}\right)}\right)$ and asymptotics $\lim_{n\to\infty} O_n/(2\log n)=1$. The upper bound follows from a maximal inradius bound $r_n=\operatorname{arccosh}\left(\frac{1}{\sin(\frac{\pi}{n})}\right)$, a cut-locus based cell decomposition, and a doubling argument that yields a pants-like structure on the doubled polygon; the lower bound is obtained by detailed hyperbolic-trigonometric analysis of the regular ideal polygon $P_n$. These results parallel Bers-type bounds for closed surfaces and illuminate the geometry of ideal polygon moduli spaces, including growth constants set by $O_n$.

Abstract

This article gives a short proof that all ideal polygons admit a short orthogeodesic decomposition. Specifically, all $n$-gons admit an orthogeodesic decomposition with orthogeodesics all of length at most $\sim 2 \log(n)$, and this is roughly optimal.

Figures (4)

• Figure 1: The different types of complementary regions to an orthogeodesic decomposition
• Figure 2: Computing the inradius of $P_n$
• Figure 3: Constructing a decomposition
• Figure 4: Computing orthogeodesic lengths on $P_n$

Theorems & Definitions (12)

• Theorem 1.1
• Proposition 2.1
• Proposition 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3
• proof