Finding Closed Quasigeodesics on Convex Polyhedra

TL;DR

This work resolves the long-standing open problem of constructively finding a closed quasigeodesic on a convex polyhedron by presenting the first finite algorithm with provable bounds. The method centers on a Real RAM traversal of a state-graph whose edges encode feasible quasigeodesic progress and on controlled unfolding arguments to bound the number of face traversals; the authors prove a pseudopolynomial upper bound on the total number of face visits and provide a near-linear-time result on the Real RAM. To bridge theory with practical finite-precision computation, they introduce the expression RAM, enabling rigorous cost analyses of algorithms manipulating radical expressions and showing how to translate the Real RAM algorithm to the Word RAM with acceptable slowdown. The paper also develops a comprehensive cost framework for radical-expression computations, including a recursive and simple cost model, and demonstrates how to extend these results to multiple input representations of polyhedra via Alexandrov gluing. Overall, the work delivers a foundational, verifiable algorithm for a classical geometric problem and a versatile computational model for exact geometric computation.

Abstract

A closed quasigeodesic is a closed curve on the surface of a polyhedron with at most $180^\circ$ of surface on both sides at all points; such curves can be locally unfolded straight. In 1949, Pogorelov proved that every convex polyhedron has at least three (non-self-intersecting) closed quasigeodesics, but the proof relies on a nonconstructive topological argument. We present the first finite algorithm to find a closed quasigeodesic on a given convex polyhedron, which is the first positive progress on a 1990 open problem by O'Rourke and Wyman. The algorithm also establishes a pseudopolynomial upper bound on the total number of visits to faces (number of line segments), namely, $O\left(\frac{n \, L^2}{ε^2 \, \ell^2}\right)$ where $n$ is the number of vertices of the polyhedron, $ε$ is the minimum curvature of a vertex, $L$ is the length of the longest edge, and $\ell$ is the smallest distance within a face between a vertex and a nonincident edge (minimum feature size of any face). On the real RAM, the algorithm's running time is also pseudopolynomial, namely $O\left(\frac{L^2}{ε^2 \, \ell^2} \, n \lg n\right)$. On a word RAM, the running time grows to $O\left(\frac{b^2 \, Δ^{36} \, L^{146}}{ε^{98} \, \ell^{146}} \, n \lg n \cdot 2^{O(|R|)}\right)$, where $Δ\leq n$ is the polyhedron's maximum vertex degree, assuming the polyhedron's intrinsic geometry is given by constant-size radical expressions with $b$-bit integers and at most $|R|$ distinct square-roots. Along the way, we introduce the expression RAM model of computation, formalizing a connection between the real RAM and word RAM hinted at by past work on exact geometric computation.

Figures (11)

• Figure 1: At a vertex of curvature $\kappa$, there is a $\kappa$-size interval of angles in which a segment of a quasigeodesic can be extended: the segment of geodesic starting on the left can continue straight in either of the pictured unfoldings, or any of the intermediate unfoldings in which the right pentagon touches only at a vertex.
• Figure 2: A source unfolding from vertex $V$ of a six-vertex polyhedron (the convex hull of a square-based pyramid and vertex $V$ which is slightly outside the pyramid), similar to Agarwal-Aronov-O'Rourke-Schevon-1997 and Demaine-O'Rourke-2007. No closed quasigeodesic can be formed by two shortest paths from $V$ to another point $P$, but there is a (vertical) closed quasigeodesic passing through $V$.
• Figure 3: Non-self-intersecting quasigeodesics may cross a face many times. For example, a $1\times 1\times L$ rectangular prism admits closed quasigeodesics which cross a face $\Omega(L)$ times.
• Figure 4: We construct a directed graph where vertices are pairs $(U, A)$, where $U$ is a polyhedron vertex and $A$ is an interval of directions/angles leaving $U$. The left figure shows two polyhedron vertices, where the space of directions leaving each vertex is partitioned into intervals of size $\leq \varepsilon/2$. The right figure shows what an edge from $(U, A)$ to $(V, B)$ represents: there exists a geodesic (purple) leaving $U$ in a direction from interval $A$ (dark green) that hits $V$ such that a quasigeodesic may continue from $V$ along any direction in interval $B$ (dark green). The lighter green interval of directions from $V$ containing $B$ shows the full range of angles that the ray may continue while being quasigeodesic.
• Figure 5: A segment of a geodesic is a straight line in the unfolding of the sequence of faces through which it passes, as in this unfolding of a regular dodecahedron.

Theorems & Definitions (64)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.6
• proof