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Ipelets for the Convex Polygonal Geometry

Nithin Parepally, Ainesh Chatterjee, Auguste Gezalyan, Hongyang Du, Sukrit Mangla, Kenny Wu, Sarah Hwang, David Mount

TL;DR

The paper addresses the need for interactive visualization tools for convex polygonal geometry by providing a library of Lua-based Ipelets for the Ipe editor. It introduces a collection of Ipelets that construct and manipulate polygonal geometric objects including Macbeath regions, Funk and Hilbert metric balls, polar bodies, and Funk/Hilbert MSTs, alongside Boolean operations, the Minkowski sum, and the minimum enclosing ball. These contributions enable researchers to build, visualize, and analyze complex convex-geometric structures directly within a familiar drawing environment, with the resources freely available. The work supports deeper geometric intuition and practical exploration in areas such as convex approximation, clustering, graph embeddings, and motion planning, by providing concrete, interactive tools for polygonal geometry.

Abstract

There are many structures, both classical and modern, involving convex polygonal geometries whose deeper understanding would be facilitated through interactive visualizations. The Ipe extensible drawing editor, developed by Otfried Cheong, is a widely used software system for generating geometric figures. One of its features is the capability to extend its functionality through programs called Ipelets. In this media submission, we showcase a collection of new Ipelets that construct a variety of geometric objects based on polygonal geometries. These include Macbeath regions, metric balls in the forward and reverse Funk distance, metric balls in the Hilbert metric, polar bodies, the minimum enclosing ball of a point set, and minimum spanning trees in both the Funk and Hilbert metrics. We also include a number of utilities on convex polygons, including union, intersection, subtraction, and Minkowski sum (previously implemented as a CGAL Ipelet). All of our Ipelets are programmed in Lua and are freely available.

Ipelets for the Convex Polygonal Geometry

TL;DR

The paper addresses the need for interactive visualization tools for convex polygonal geometry by providing a library of Lua-based Ipelets for the Ipe editor. It introduces a collection of Ipelets that construct and manipulate polygonal geometric objects including Macbeath regions, Funk and Hilbert metric balls, polar bodies, and Funk/Hilbert MSTs, alongside Boolean operations, the Minkowski sum, and the minimum enclosing ball. These contributions enable researchers to build, visualize, and analyze complex convex-geometric structures directly within a familiar drawing environment, with the resources freely available. The work supports deeper geometric intuition and practical exploration in areas such as convex approximation, clustering, graph embeddings, and motion planning, by providing concrete, interactive tools for polygonal geometry.

Abstract

There are many structures, both classical and modern, involving convex polygonal geometries whose deeper understanding would be facilitated through interactive visualizations. The Ipe extensible drawing editor, developed by Otfried Cheong, is a widely used software system for generating geometric figures. One of its features is the capability to extend its functionality through programs called Ipelets. In this media submission, we showcase a collection of new Ipelets that construct a variety of geometric objects based on polygonal geometries. These include Macbeath regions, metric balls in the forward and reverse Funk distance, metric balls in the Hilbert metric, polar bodies, the minimum enclosing ball of a point set, and minimum spanning trees in both the Funk and Hilbert metrics. We also include a number of utilities on convex polygons, including union, intersection, subtraction, and Minkowski sum (previously implemented as a CGAL Ipelet). All of our Ipelets are programmed in Lua and are freely available.
Paper Structure (10 sections, 6 equations, 8 figures)

This paper contains 10 sections, 6 equations, 8 figures.

Table of Contents

  1. Structures
  2. Macbeath Regions
  3. Funk and Hilbert Balls
  4. Polar Bodies
  5. Funk and Hilbert Minimum Spanning Trees
  6. Operations
  7. Boolean Operations
  8. Minkowski Sum
  9. Minimum Enclosing Ball
  10. Installation

Figures (8)

  • Figure 1: (a) A Macbeath region and (b) a collection of Macbeath regions
  • Figure 2: (a) The Funk metric, (b) the reverse Funk metric, and (c) the Hilbert metric.
  • Figure 3: (a) Funk, (b) reverse Funk, and (c) Hilbert balls (with and without spokes).
  • Figure 4: The polar body.
  • Figure 5: (a) The Hilbert MST and (b) the Funk MST
  • ...and 3 more figures

Theorems & Definitions (6)

  • Definition 1: Macbeath Region
  • Definition 2: (Forward) Funk Metric
  • Definition 3: Reverse Funk Metric
  • Definition 4: Hilbert metric
  • Definition 5: Polar Body
  • Definition 6: Minkowski Sum