French Onion Soup, Ipelets for Points and Polygons
Klint Faber, Auguste H. Gezalyan, Adam Martinson, Aniruddh Mutnuru, Nithin Parepally, Ryan Parker, Mihil Sreenilayam, Aram Zaprosyan, David M. Mount
TL;DR
The work addresses the need for interactive, visual exploration of geometric structures on 2D point sets and polygons. It introduces a suite of Lua-based Ipelets that extend the Ipe drawing editor to construct and visualize quadtrees, trapezoidal maps with $O(\log n)$ point location, onion decompositions, beta skeletons, floating bodies (Dupin and convex), polygon triangulations and sampling, and Sierpiński fractals within simple polygons. The contributions include freely available Ipelets with array export for easy reuse, along with clear installation guidance. This facilitates education and rapid experimentation in computational geometry, GIS, and related visualization tasks.
Abstract
There are many structures, both classical and modern, involving point-sets and polygons whose deeper understanding can 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 based structures based on point sets and polygons. These include quadtrees, trapezoidal maps, beta skeletons, floating bodies of convex polygons, onion graphs, fractals (Sierpiński triangle and carpet), simple polygon triangulations, and random point sets in simple polygons. All of our Ipelets are programmed in Lua and are freely available.