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Curves of Finite Total Curvature

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Discrete Differential Geometry

Part of the book series: Oberwolfach Seminars ((OWS,volume 38))

Abstract

We consider the class of curves of finite total curvature, as introduced by Milnor. This is a natural class for variational problems and geometric knot theory, and since it includes both smooth and polygonal curves, its study shows us connections between discrete and differential geometry. To explore these ideas, we consider theorems of Fáry/Milnor, Schur, Chakerian and Wienholtz.

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References

  1. Stephanie B. Alexander and Richard L. Bishop, The Fáry-Milnor theorem in Hadamard manifolds, Proc. Amer. Math. Soc. 126:11 (1998), 3427–3436.

    Article  MATH  MathSciNet  Google Scholar 

  2. Sahbi Ayari and Serge Dubuc, La formule de Cauchy sur la longueur d’une courbe, Canad. Math. Bull. 40:1 (1997), 3–9.

    MATH  MathSciNet  Google Scholar 

  3. Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Clarendon/Oxford, 2000.

    Google Scholar 

  4. Helmut Alt and Michael Godau, Computing the Fréchet distance between two polygonal curves, Internat. J. Comput. Geom. Appl. 5:1–2 (1995), 75–91, Proc. 8th ACM Symp. Comp. Geom. (Berlin, 1992).

    Article  MATH  MathSciNet  Google Scholar 

  5. Aleksandr D. Alexandrov and Yuri G. Reshetnyak, General theory of irregular curves, Math. Appl. (Soviet) 29, Kluwer, Dordrecht, 1989.

    Google Scholar 

  6. Louis I. Alpert and Leopoldo V. Toralballa, An elementary definition of surface area in \( \mathbb{E}^{n + 1} \) for smooth surfaces, Pacific J. Math. 40 (1972), 261–268.

    MATH  MathSciNet  Google Scholar 

  7. Martin Aigner and Günter M. Ziegler, Proofs from The Book, Springer, 1998.

    Google Scholar 

  8. Sterling K. Berberian, Fundamentals of real analysis, Springer, 1998.

    Google Scholar 

  9. Frederick Brickell and Chuan-Chih Hsiung, The total absolute curvature of closed curves in Riemannian manifolds, J. Differential Geometry 9 (1974), 177–193.

    MATH  MathSciNet  Google Scholar 

  10. Wilhelm Blaschke, Ungleichheiten von H. A. Schwarz und A. Schur für Raumkurven mit vorgeschriebener Krümmung., Hamb. Abh. 1 (1921), 49–53.

    Google Scholar 

  11. _____, Vorlesungen über Differentialgeometrie, Springer, Berlin, 1924.

    Google Scholar 

  12. Ralph P. Boas, Jr., A primer of real functions, fourth ed., Carus Math. Manus., no. 13, Math. Assoc. Amer., 1996.

    Google Scholar 

  13. Karol Borsuk, Sur la courbure totale des courbes fermées, Ann. Soc. Polon. Math. 20 (1947), 251–265 (1948).

    MATH  MathSciNet  Google Scholar 

  14. Gregory R. Buck and Jonathon K. Simon, Thickness and crossing number of knots, Topol. Appl. 91:3 (1999), 245–257.

    Article  MATH  MathSciNet  Google Scholar 

  15. Georges-Louis Leclerc, Comte de Buffon, Essai d’arithmétique morale, Histoire naturelle, générale er particulière, Suppl. 4, 1777, pp. 46–123; www.buffon.cnrs.fr/ice/ice_book_detail-fr-text-buffon-buffon_hn-33-7.html.

    Google Scholar 

  16. Constantin Carathéodory, Vorlesungen über reele Funktionen, Teubner, 1918, reprinted 2004 by AMS/Chelsea.

    Google Scholar 

  17. Augustin-Louis Cauchy, Deuxième mémoire sur les polygones et les polyèdres, J. École Polytechnique 16 (1813), 87–98.

    Google Scholar 

  18. _____, Notes sur divers théorèmes relatifs à la rectification des courbes, et à la quadrature des surfaces, C. R. Acad. Sci. Paris 13 (1841), 1060–1063, reprinted in Oeuvres complètes 6, Gauthier-Villars, 1888, pp. 369–375.

    Google Scholar 

  19. Lamberto Cesari, Surface area, Ann. Math. Stud., no. 35, Princeton, 1956.

    Google Scholar 

  20. _____, Surface area, Global Differential Geometry (S.S. Chern, ed.), Math. Assoc. Amer., 1989, pp. 270–302.

    Google Scholar 

  21. Jason Cantarella, Joseph H.G. Fu, Robert B. Kusner, John M. Sullivan, and Nancy C. Wrinkle, Criticality for the Gehring link problem, Geometry and Topology 10 (2006), 2055–2115; arXiv.org/math.DG/0402212.

    Article  MATH  MathSciNet  Google Scholar 

  22. Gulbank D. Chakerian, An inequality for closed space curves, Pacific J. Math. 12 (1962), 53–57.

    MATH  MathSciNet  Google Scholar 

  23. _____, On some geometric inequalities, Proc. Amer. Math. Soc. 15 (1964), 886–888.

    Article  MATH  MathSciNet  Google Scholar 

  24. Shiing Shen Chern, Curves and surfaces in euclidean space, Global Differential Geometry (S.S. Chern, ed.), Math. Assoc. Amer., 1989, pp. 99–139.

    Google Scholar 

  25. Jason Cantarella, Greg Kuperberg, Robert B. Kusner, and John M. Sullivan, The second hull of a knotted curve, Amer. J. Math 125 (2003), 1335–1348; arXiv.org/math.GT/0204106.

    Article  MATH  MathSciNet  Google Scholar 

  26. Jason Cantarella, Robert B. Kusner, and John M. Sullivan, On the minimum ropelength of knots and links, Invent. Math. 150 (2002), 257–286; arXiv.org/math.GT/0103224.

    Article  MATH  MathSciNet  Google Scholar 

  27. Robert Connelly, Rigidity and energy, Invent. Math. 66 (1982), 11–33.

    Article  MATH  MathSciNet  Google Scholar 

  28. Morgan W. Crofton, On the theory of local probability, Phil. Trans. R. Soc. London 158 (1868), 181–199.

    Article  Google Scholar 

  29. Peter Cromwell, Polyhedra, Cambridge, 1997.

    Google Scholar 

  30. Manfredo P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, 1976.

    Google Scholar 

  31. Elizabeth Denne, Yuanan Diao, and John M. Sullivan, Quadrisecants give new lower bounds for the ropelength of a knot, Geometry and Topology 10 (2006), 1–26; arXiv. org/math.DG/0408026.

    Article  MATH  MathSciNet  Google Scholar 

  32. Elizabeth Denne, Alternating quadrisecants of knots, Ph.D. thesis, Univ. Illinois, Urbana, 2004, arXiv.org/math.GT/0510561.

    Google Scholar 

  33. Elizabeth Denne and John M. Sullivan, The distortion of a knotted curve, preprint, 2004, arXiv.org/math.GT/0409438v1.

    Google Scholar 

  34. _____, Convergence and isotopy type for graphs of finite total curvature, Discrete Differential Geometry (A.I. Bobenko, P. Schröder, J.M. Sullivan, G.M. Ziegler, eds.), Oberwolfach Seminars, vol. 38, Birkhäuser, 2008, this volume, pp. 163–174; arXiv.org/math.GT/0606008.

    Google Scholar 

  35. István Fáry, Sur la courbure totale d’une courbe gauche faisant un noe ud, Bull. Soc. Math. France 77 (1949), 128–138.

    MATH  MathSciNet  Google Scholar 

  36. Werner Fenchel, Über Krümmung und Windung geschlossener Raumkurven, Math. Ann. 101 (1929), 238–252; www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN235181684_0101.

    Article  MathSciNet  Google Scholar 

  37. Werner Fenchel, On the differential geometry of closed space curves, Bull. Amer. Math. Soc. 57 (1951), 44–54.

    Article  MATH  MathSciNet  Google Scholar 

  38. Maurice Fréchet, Sur l’écart de deux courbes et sur les courbes limites, Trans. Amer. Math. Soc. 6 (1905), 435–449.

    Article  MathSciNet  Google Scholar 

  39. Casper Goffman and George Pedrick, First course in functional analysis, second ed., Chelsea, 1983.

    Google Scholar 

  40. Lawrence M. Graves, Theory of functions of real variables, McGraw Hill, 1946.

    Google Scholar 

  41. Mikhael Gromov, Homotopical effects of dilatation, J. Diff. Geom. 13 (1978), 303–310.

    MATH  MathSciNet  Google Scholar 

  42. _____, Structures métriques pour les variétés riemanniennes, Cedic, Paris, 1981, edited by J. Lafontaine and P. Pansu.

    Google Scholar 

  43. _____, Filling Riemannian manifolds, J. Diff. Geom. 18 (1983), 1–147.

    MATH  MathSciNet  Google Scholar 

  44. Tim Hoffmann and Nadja Kutz, Discrete curves in ℂP1 and the Toda lattice, Stud. Appl. Math. 113:1 (2004), 31–55; arXiv.org/math.DG/0208190.

    Article  MathSciNet  Google Scholar 

  45. Tim Hoffmann, Discrete Hashimoto surfaces and a doubly discrete smoke-ring flow, Discrete Differential Geometry (A.I. Bobenko, P. Schröder, J.M. Sullivan, G.M. Ziegler, eds.), Oberwolfach Seminars, vol. 38, Birkhäuser, 2008, this volume, pp. 95–115; arXiv.org/math.DG/0007150.

    Google Scholar 

  46. Roger A. Horn, On Fenchel’s theorem, Amer. Math. Monthly 78 (1971), 380–381.

    Article  MATH  MathSciNet  Google Scholar 

  47. Camille Jordan, Cours d’analyse de l’école polytechnique, Gauthier-Villars, 1893.

    Google Scholar 

  48. Daniel A. Klain and Gian-Carlo Rota, Introduction to geometric probability, Cambridge, 1997.

    Google Scholar 

  49. Robert B. Kusner and John M. Sullivan, On distortion and thickness of knots, Topology and Geometry in Polymer Science (Whittington, Sumners, and Lodge, eds.), IMA Vol. 103, Springer, 1997, pp. 67–78; arXiv.org/dg-ga/9702001.

    Google Scholar 

  50. Henri Lebesgue, Intégrale, longuer, aire, Annali di Mat. pura appl. 7 (1902), 231–359.

    Google Scholar 

  51. Heinrich Liebmann, Elementarer Beweis des Fenchelschen Satzes über die Krümmung geschlossener Raumkurven, Sitz.ber. Akad. Berlin (1929), 392–393.

    Google Scholar 

  52. Jiří Matoušek, Using the Borsuk-Ulam theorem, Springer, Berlin, 2003.

    Google Scholar 

  53. John W. Milnor, On the total curvature of knots, Ann. of Math. 52 (1950), 248–257.

    Article  MathSciNet  Google Scholar 

  54. Marston Morse, A special parametrization of curves, Bull. Amer. Math. Soc. 42 (1936), 915–922.

    Article  MATH  Google Scholar 

  55. Frank Morgan, Geometric measure theory: A beginner’s guide, Academic Press, 1988.

    Google Scholar 

  56. Joseph O’Rourke, On the development of the intersection of a plane with a polytope, Comput. Geom. Theory Appl. 24:1 (2003), 3–10; arXiv.org/cs.CG/0006035v3.

    MATH  MathSciNet  Google Scholar 

  57. Luis A. Santaló, Integral geometry, Global Differential Geometry, Math. Assoc. Amer., 1989, pp. 303–350.

    Google Scholar 

  58. _____, Integral geometry and geometric probability, second ed., Cambridge, 2004.

    Google Scholar 

  59. Ludwig Scheeffer, Allgemeine Untersuchungen über Rectification der Curven, Acta Math. 5 (1884–85), 49–82.

    Article  MathSciNet  Google Scholar 

  60. Hermann Amandus Schwarz, Sur une définition erronée de l’aire d’une surface courbe, Ges. math. Abhandl., vol. 2, Springer, 1890, pp. 309–311 and 369–370.

    Google Scholar 

  61. Axel Schur, Über die Schwarzsche Extremaleigenschaft des Kreises unter den Kurven konstanter Krümmung, Math. Annalen 83 (1921), 143–148; www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN235181684_0083.

    Article  MathSciNet  Google Scholar 

  62. Erhard Schmidt, Über das Extremum der Bogenlänge einer Raumkurve bei vorgeschriebenen Einschränkungen ihrer Krümmung, Sitz.ber. Akad. Berlin (1925), 485–490.

    Google Scholar 

  63. Carsten Schmitz, The theorem of Fáry and Milnor for Hadamard manifolds, Geom. Dedicata 71:1 (1998), 83–90.

    Article  MATH  MathSciNet  Google Scholar 

  64. Joseph-Alfred Serret, Cours de calcul différentiel et intégral, vol. 2, Gauthier-Villars, Paris, 1868; historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05270001&seq=311.

    Google Scholar 

  65. John M. Sullivan, Curvatures of smooth and discrete surfaces, Discrete Differential Geometry (A.I. Bobenko, P. Schröder, J.M. Sullivan, G.M. Ziegler, eds.), Oberwolfach Seminars, vol. 38, Birkhäuser, 2008, this volume, pp. 175–188; arXiv.org/0710.4497.

    Google Scholar 

  66. Kouki Taniyama, Total curvature of graphs in Euclidean spaces, Differential Geom. Appl. 8:2 (1998), 135–155.

    Article  MathSciNet  Google Scholar 

  67. Leonida Tonelli, Fondamenti di calcolo delle variazione, Zanichelli, Bologna, 1921.

    Google Scholar 

  68. Konrad Voss, Eine Bemerkung über die Totalkrümmung geschlossener Raumkurven, Arch. Math. 6 (1955), 259–263.

    Article  MATH  MathSciNet  Google Scholar 

  69. Max Wardetzky, Convergence of the cotangent formula: An overview, Discrete Differential Geometry (A.I. Bobenko, P. Schröder, J.M. Sullivan, G.M. Ziegler, eds.), Oberwolfach Seminars, vol. 38, Birkhäuser, 2008, this volume, pp. 275–286.

    Google Scholar 

  70. William P. Ziemer, Weakly differentiable functions, GTM 120, Springer, 1989.

    Google Scholar 

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Authors and Affiliations

  1. Institut für Mathematik, MA 3-2, Technische Universität Berlin, Str. des 17. Juni 136, 10623, Berlin, Germany

    John M. Sullivan

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  1. John M. Sullivan
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Editors and Affiliations

  1. Institut für Mathematik, MA 8-3, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Alexander I. Bobenko

  2. Institut für Mathematik, MA 3-2, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    John M. Sullivan

  3. Department of Computer Science, Caltech, MS 256-80, 1200 E. California Blvd., Pasadena, CA, 91125, USA

    Peter Schröder

  4. Institut für Mathematik, MA 6-2, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Günter M. Ziegler

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Sullivan, J.M. (2008). Curves of Finite Total Curvature. In: Bobenko, A.I., Sullivan, J.M., Schröder, P., Ziegler, G.M. (eds) Discrete Differential Geometry. Oberwolfach Seminars, vol 38. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8621-4_7

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