Document Open Access Logo

Finding Closed Quasigeodesics on Convex Polyhedra

Authors Erik D. Demaine, Adam C. Hesterberg, Jason S. Ku

Thumbnail PDF


  • Filesize: 0.66 MB
  • 13 pages

Document Identifiers

Author Details

Erik D. Demaine
  • Computer Science and Artificial Intelligence Laboratory, MIT, Cambridge, MA, USA
Adam C. Hesterberg
  • Computer Science and Artificial Intelligence Laboratory, MIT, Cambridge, MA, USA
Jason S. Ku
  • Department of Electrical Engineering and Computer Science, MIT, Cambridge, MA, USA


The authors thank Zachary Abel, Nadia Benbernou, Fae Charlton, Jayson Lynch, Joseph O'Rourke, Diane Souvaine, and David Stalfa for discussions related to this paper.

Cite AsGet BibTex

Erik D. Demaine, Adam C. Hesterberg, and Jason S. Ku. Finding Closed Quasigeodesics on Convex Polyhedra. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 33:1-33:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


A closed quasigeodesic is a closed loop on the surface of a polyhedron with at most 180° of surface on both sides at all points; such loops can be locally unfolded straight. In 1949, Pogorelov proved that every convex polyhedron has at least three (non-self-intersecting) closed quasigeodesics, but the proof relies on a nonconstructive topological argument. We present the first finite algorithm to find a closed quasigeodesic on a given convex polyhedron, which is the first positive progress on a 1990 open problem by O'Rourke and Wyman. The algorithm’s running time is pseudopolynomial, namely O(n²/ε² L/𝓁 b) time, where ε is the minimum curvature of a vertex, L is the length of the longest edge, 𝓁 is the smallest distance within a face between a vertex and a nonincident edge (minimum feature size of any face), and b is the maximum number of bits of an integer in a constant-size radical expression of a real number representing the polyhedron. We take special care in the model of computation and needed precision, showing that we can achieve the stated running time on a pointer machine supporting constant-time w-bit arithmetic operations where w = Ω(lg b).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • polyhedra
  • geodesic
  • pseudopolynomial
  • geometric precision


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Hans Werner Ballmann. Der Satz von Lusternik und Schnirelmann. Bonner Mathematische Schriften, 102:1-25, 1978. Google Scholar
  2. Werner Ballmann, Gudlaugur Thorbergsson, and Wolfgang Ziller. On the existence of short closed geodesics and their stability properties. In Seminar on Minimal Submanifolds, pages 53-63. Princeton University Press, 1983. URL:
  3. Amir M. Ben-Amram. What is a "pointer machine"? SIGACT News, 26(2):88-95, June 1995. URL:
  4. A. Bertoni, G. Mauri, and N. Sabadini. Simulations among classes of random access machines and equivalence among numbers succinctly represented. Annals of Discrete Mathematics, 25:65-90, 1985. Google Scholar
  5. George D. Birkhoff. Dynamical Systems, volume 9 of Colloquium Publications. American Mathematical Society, 1927. URL:
  6. Christoph Burnikel, Stefan Funke, Kurt Mehlhorn, Stefan Schirra, and Susanne Schmitt. A separation bound for real algebraic expressions. In Proceedings of the 9th Annual European Symposium on Algorithms, volume 2161 of Lecture Notes in Computer Science, pages 254-265, Aarhus, Denmark, August 2001. URL:
  7. Timothy M. Chan and Mihai Pǎtraşcu. Transdichotomous results in computational geometry, I: Point location in sublogarithmic time. SIAM Journal on Computing, 39(2):703-729, 2009. URL:
  8. Timothy M. Chan and Mihai Pǎtraşcu. Transdichotomous results in computational geometry, II: Offline search, 2010. Originally published at STOC 2007. URL:
  9. Erik D. Demaine and Joseph O'Rourke. Geodesics: Lyusternik-Schnirelmann. In Geometric Folding Algorithms: Linkages, Origami, Polyhedra, section 24.4, pages 372-375. Cambridge University Press, Cambridge, 2007. Google Scholar
  10. James R. Driscoll, Neil Sarnak, Daniel D. Sleator, and Robert E. Tarjan. Making data structures persistent. Journal of Computer and System Sciences, 38(1):86-124, 1989. Google Scholar
  11. Michael L. Fredman and Dan E. Willard. Surpassing the information theoretic bound with fusion trees. Journal of Computer and System Sciences, 47(3):424-436, 1993. URL:
  12. David Harvey and Joris van der Hoeven. Integer multiplication in time O(n log n). HAL Preprint hal-02070778, 2019. URL:
  13. Giuseppe F. Italiano and Rajeev Raman. Topics in data structures. In Mikhail J. Atallah and Marina Blanton, editors, Algorithms and Theory of Computation Handbook, volume 1, chapter 5, pages 5-1 - 5-29. CRC Press, second edition, 2010. Google Scholar
  14. Jin-Ichi Itoh, Joël Rouyer, and Costin Vîlcu. Polyhedra with simple dense geodesics. Differential Geometry and its Applications, 66:242-252, 2019. URL:
  15. Daniel Kane, Gregory N. Price, and Erik D. Demaine. A pseudopolynomial algorithm for Alexandrov’s Theorem. In Proceedings of the 11th Algorithms and Data Structures Symposium, volume 5664 of Lecture Notes in Computer Science, pages 435-446, Banff, Canada, August 2009. Google Scholar
  16. Donald E. Knuth. The Art of Computer Programming, volume 2. Addison-Wesley, 1969. Google Scholar
  17. Lazar Lyusternik and Lev Schnirelmann. Sur le probléme de trois géodésiques fermées sur les surfaces de genre 0. Comptes Rendus de l'Académie des Sciences de Paris, 189:269-271, 1929. Google Scholar
  18. Joseph O'Rourke. Personal communication, 2020. Google Scholar
  19. Aleksei Vasilevich Pogorelov. Quasi-geodesic lines on a convex surface. Matematicheskii Sbornik, 25(62):275-306, 1949. English translation in American Mathematical Society Translations 74, 1952. Google Scholar
  20. Henri Poincaré. Sur les lignes géodésiques des surfaces convexes. Transactions of the American Mathematical Society, 6(3):237-274, 1905. Google Scholar
  21. Arnold Schönhage. On the power of random access machines. In Proceedings of the 6th International Colloquium on Automata, Languages, and Programming, volume 71 of Lecture Notes in Computer Science, pages 520-529, 1979. Google Scholar
  22. Michael Ian Shamos. Computational Geometry. PhD thesis, Yale University, 1978. Google Scholar
  23. Vikram Sharma and Chee K. Yap. Robust geometric computation. In Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Tóth, editors, Handbook of Discrete and Computational Geometry, chapter 45, pages 1189-1223. CRC Press, third edition, 2018. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail