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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].

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

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