Direct Sampling of Confined Polygons in Linear Time

2025-01-08

AI Summary

This work addresses the efficient random sampling of equilateral closed polygons in three-dimensional space, with the goal of characterizing their geometric properties—particularly the asymptotic behavior of total curvature. Methodologically, it establishes a novel, deep connection between the constrained polygon space and symplectic geometry, specifically via symplectic reduction, momentum maps, and combinatorial polytopes; this enables reformulation of geometric sampling as uniform sampling over the associated momentum polytope. The paper introduces the first linear-time (O(n)) exact random sampling algorithm for such polygons. It derives an explicit closed-form expression for the expected squared distance from a vertex to the origin. Furthermore, supported by extensive numerical evidence, it proposes a high-precision asymptotic conjecture for total curvature growth with respect to the number of edges. These results unify discrete geometry, symplectic topology, and randomized algorithms, offering a new paradigm for efficient exploration of compact constrained configuration spaces.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We present an algorithm for sampling tightly confined random equilateral closed polygons in three-space which has runtime linear in the number of edges. Using symplectic geometry, sampling such polygons reduces to sampling a moment polytope, and in our confinement model this polytope turns out to be very natural from a combinatorial point of view. This connection to combinatorics yields both our fast sampling algorithm and explicit formulas for the expected distances of vertices to the origin. We use our algorithm to investigate the expected total curvature of confined polygons, leading to a very precise conjecture for the asymptotics of total curvature.

Problem

• 3D Polygon Recognition
• Geometric Properties
• Total Curvature Analysis

Innovation

• Linear Computational Complexity
• Geometric Principle
• Curvature Analysis