The Convex Hull of Two Circles in \(\mathbb{R}^{3}\)

  • Evan D. NashEmail author
  • Ata Firat Pir
  • Frank Sottile
  • Li Ying
Part of the Fields Institute Communications book series (FIC, volume 80)


We describe convex hulls of the simplest compact space curves, reducible quartics consisting of two circles. When the circles do not meet in complex projective space, their algebraic boundary contains an irrational ruled surface of degree eight whose ruling forms a genus one curve. We classify which curves arise, classify the face lattices of the convex hulls, and determine which are spectrahedra. We also discuss an approach to these convex hulls using projective duality.

MSC 2010 codes:

52A05 14P10 90C22 



This article was initiated during the Apprenticeship Weeks (22 August–2 September 2016), led by Bernd Sturmfels, as part of the Combinatorial Algebraic Geometry Semester at the Fields Institute. Sottile, Pir, and Ying were supported in part by the National Science Foundation grant DMS-1501370.


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Copyright information

© Springer Science+Business Media LLC 2017

Authors and Affiliations

  • Evan D. Nash
    • 1
    Email author
  • Ata Firat Pir
    • 2
  • Frank Sottile
    • 2
  • Li Ying
    • 2
  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA

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