Kubjas, Kaie; Parrilo, Pablo A.; Sturmfels, Bernd How to flatten a soccer ball. (English) Zbl 1406.14042 Conca, Aldo (ed.) et al., Homological and computational methods in commutative algebra. Dedicated to Winfried Bruns on the occasion of his 70th birthday. Proceedings of the INdAM conference, Cortona, Italy, May 30 – June 3, 2016. Cham: Springer (ISBN 978-3-319-61942-2/hbk; 978-3-319-61943-9/ebook). Springer INdAM Series 20, 141-162 (2017). Summary: 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.For the entire collection see [Zbl 1387.13002]. Cited in 4 Documents MSC: 14Q10 Computational aspects of algebraic surfaces 14P10 Semialgebraic sets and related spaces 65D18 Numerical aspects of computer graphics, image analysis, and computational geometry 90C25 Convex programming 90C90 Applications of mathematical programming Keywords:cylindrical algebraic decomposition; polynomial map; semialgebraic set Software:Macaulay2 PDFBibTeX XMLCite \textit{K. Kubjas} et al., Springer INdAM Ser. 20, 141--162 (2017; Zbl 1406.14042) Full Text: DOI arXiv