# Packing Ovals in Optimized Regular Polygons

## TL;DR

This paper introduces a numerical method for packing convex ovals into optimized regular polygons, demonstrating effective solutions for complex packing problems with high accuracy.

## Contribution

It develops a novel framework combining Lagrange multipliers and nonlinear optimization for packing convex ovals into optimized polygons.

## Key findings

- Achieved tight packings for 4 to 10 ovals in polygons with 3 to 10 sides.
- Successfully solved 224 diverse packing problems.
- Method provides credible solutions for complex convex packing scenarios.

## Abstract

We present a model development framework and numerical solution approach to the general problem-class of packing convex objects into optimized convex containers. Specifically, here we discuss the problem of packing ovals (egg-shaped objects, defined here as generalized ellipses) into optimized regular polygons in $\mathbb{R}^2$. Our solution strategy is based on the use of embedded Lagrange multipliers, followed by nonlinear (global-local) optimization. The numerical results are attained using randomized starting solutions refined by a single call to a local optimization solver. We obtain credible, tight packings for packing 4 to 10 ovals into regular polygons with 3 to 10 sides in all (224) test problems presented here, and for other similarly difficult packing problems.

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Source: https://tomesphere.com/paper/1901.07056