How to flatten a soccer ball

TL;DR

This paper investigates the geometric transformation of semialgebraic sets in 3D under polynomial maps, revealing complex boundary structures and exploring computational methods like cylindrical algebraic decompositions.

Contribution

It provides a detailed analysis of the boundary curves of polynomial images of semialgebraic sets, introducing new techniques for their computation and connections to convex optimization.

Findings

Boundary of the image is given by two highly singular curves.

Explicit determination of these boundary curves.

Connections established between algebraic geometry and convex optimization.

Abstract

This is an experimental case study in real algebraic geometry, aimed at computing the image of a semialgebraic subset of 3-space under a polynomial map into the plane. For general instances, the boundary of the image is given by two highly singular curves. We determine these curves and show how they demarcate the "flattened soccer ball". We explore cylindrical algebraic decompositions, by working through concrete examples. Maps onto convex polygons and connections to convex optimization are also discussed.