Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alexander Barvinok: A course in convexity, Graduate Studies in Mathematics 54, American Mathematical Society, Providence, RI, 2002.
Inversions-Technik GmbH Basle: Formerly Oloid AG, www.oloid.ch/index.php/en/.
Hans Dirnböck and Hellmuth Stachel: The development of the oloid, J. Geom. Graph. 1 (1997) 105–118.
Laura Escobar and Allen Knutson: The multidegree of the multi-image variety, in Combinatorial Algebraic Geometry, 283–296, Fields Inst. Commun. 80, Fields Inst. Res. Math. Sci., 2017.
Steven R. Finch: Convex hull of two orthogonal disks, arXiv:1211.4514 [math.MG].
Nicola Geismann, Michael Hemmer, and Elmar Schömer: The convex hull of ellipsoids, in SCG’01 Proceedings of the seventeenth annual symposium on computational geometry, 321–322, Association for Computing Machinery, New York, 2001.
_________ : The convex hull of ellipsoids 2001, youtu.be/Tq9OS5iIcBc.
Branko Grünbaum: Convex polytopes, Graduate Texts in Mathematics 221, Springer-Verlag, New York, 2003.
Robin Hartshorne: Algebraic geometry, Graduate Texts in Mathematics 52, Springer-Verlag, New York-Heidelberg, 1977.
Hiroshi Ira: The development of the two-circle-roller in a numerical way, unpublished note, 2011, ilabo.bufsiz.jp/.
Trygve Johnsen: Plane projections of a smooth space curve, in Parameter spaces (Warsaw, 1994), 89–110, Banach Center Publ. 36, Polish Acad. Sci., Warsaw, 1996
Kathlén Kohn, Bernt Ivar Utstøl Nødland, and Paolo Tripoli: Secants, bitangents, and their congruences, in Combinatorial Algebraic Geometry, 87–112, Fields Inst. Commun. 80, Fields Inst. Res. Math. Sci., 2017.
Kristian Ranestad and Bernd Sturmfels: The convex hull of a variety, in Notions of Positivity and the Geometry of Polynomials, 331–344, Trends Math., Birkhäuser/Springer Basel AG, Basel, 2011.
_________ : On the convex hull of a space curve, Adv. Geom. 12 (2012) 157–178.
Raman Sanyal, Frank Sottile, and Bernd Sturmfels: Orbitopes, Mathematika 57 (2011) 275–314.
Paul Schatz: Oloid, a device to generate a tumbling motion, Swiss Patent No. 500,000, 1929.
Claus Scheiderer: Semidefinite representation for convex hulls of real algebraic curves, SIAM J. Appl. Algebra Geom, arXiv:1208.3865 [math.AG].
_________ : Semidefinitely representable convex sets, SIAM J. Appl. Algebra Geom (in appear).
Abraham Seidenberg: A new decision method for elementary algebra, Ann. of Math. (2) 60 (1954) 365–374.
Rainer Sinn: Algebraic boundaries of \(\mathop{\mathrm{SO}}\nolimits (2)\)-orbitopes, Discrete Comput. Geom. 50 (2013) 219–235.
Bernd Sturmfels: Fitness, apprenticeship, and polynomials, in Combinatorial Algebraic Geometry, 1–19, Fields Inst. Commun. 80, Fields Inst. Res. Math. Sci., 2017.
Alfred Tarski: A Decision Method for Elementary Algebra and Geometry, RAND Corporation, Santa Monica, Calif., 1948.
Cynthia Vinzant: Edges of the Barvinok–Novik orbitope, Discrete Comput. Geom. 46 (2011) 479–487.
Günter M. Ziegler: Lectures on polytopes, Graduate Texts in Mathematics 152, Springer-Verlag, New York, 1995.
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer Science+Business Media LLC
About this chapter
Cite this chapter
Nash, E.D., Pir, A.F., Sottile, F., Ying, L. (2017). The Convex Hull of Two Circles in \(\mathbb{R}^{3}\) . In: Smith, G., Sturmfels, B. (eds) Combinatorial Algebraic Geometry. Fields Institute Communications, vol 80. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-7486-3_14
Download citation
DOI: https://doi.org/10.1007/978-1-4939-7486-3_14
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-7485-6
Online ISBN: 978-1-4939-7486-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)