How to Flatten a Soccer Ball

  • Kaie Kubjas
  • Pablo A. Parrilo
  • Bernd SturmfelsEmail author
Part of the Springer INdAM Series book series (SINDAMS, volume 20)


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.


Cylindrical algebraic decomposition Polynomial map Semialgebraic set 



Pablo Parrilo was supported by AFOSR FA9550-11-1-0305. Bernd Sturmfels was supported by NSF grant DMS-1419018 and the Einstein Foundation Berlin. Part of this work was done while the authors visited the Simons Institute for the Theory of Computing at UC Berkeley.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Kaie Kubjas
    • 1
  • Pablo A. Parrilo
    • 2
  • Bernd Sturmfels
    • 3
    Email author
  1. 1.Department of Mathematics and Systems AnalysisAalto UniversityEspooFinland
  2. 2.Laboratory for Information and Decision SystemsMassachusetts Institute of TechnologyCambridgeUSA
  3. 3.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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