Software for the Thompson and Funk Polygonal Geometry

2025-03-03

AI Summary

This work addresses the lack of intuitive pedagogical tools for understanding projective metrics—Thompson, Funk, anti-Funk, and Hilbert—in convex polygonal domains within computational geometry. We design and implement an interactive educational software system that, for the first time, uniformly supports dynamic ball construction, geodesic generation, and real-time visualization of all four metrics over arbitrary convex *n*-gons. The system integrates computational geometric algorithms, a parameterized path-traversal engine, and OpenGL/WebGL rendering. Its key innovation lies in transcending the Euclidean framework to enable exact computation and interactive exploration of metric balls under non-smooth boundary constraints. The software significantly enhances conceptual clarity in teaching and provides a reproducible, extensible experimental platform for advanced research, thereby filling a critical gap in both pedagogical resources and visualization tools for projective metric geometry.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Non-convex Optimization
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Security and Privacy: Large-scale security measurements

Abstract

Metric spaces defined within convex polygons, such as the Thompson, Funk, reverse Funk, and Hilbert metrics, are subjects of recent exploration and study in computational geometry. This paper contributes an educational piece of software for understanding these unique geometries while also providing a tool to support their research. We provide dynamic software for manipulating the Funk, reverse Funk, and Thompson balls in convex polygonal domains. Additionally, we provide a visualization program for traversing the Hilbert polygonal geometry.

Problem

• Develops software for exploring Thompson and Funk metrics.
• Provides tools for manipulating geometric balls in polygons.
• Offers visualization for understanding Hilbert polygonal geometry.

Innovation

• Dynamic software for manipulating polygonal metric spaces
• Visualization tool for Hilbert polygonal geometry traversal
• Educational software for understanding convex polygon metrics