Journal of Geometry

, 109:4 | Cite as

Monge surfaces and planar geodesic foliations



A Monge surface is a surface obtained by sweeping a generating plane curve along a trajectory that is orthogonal to the moving plane containing the curve. Locally, they are characterized as being foliated by a family of planar geodesic lines of curvature. We call surfaces with the latter property PGF surfaces, and investigate the global properties of these two naturally defined objects. The only compact orientable PGF surfaces are tori; these are globally Monge surfaces, and they have a simple characterization in terms of the directrix. We show how to produce many examples of Monge tori and Klein bottles, as well as tori that do not have a closed directrix.


Monge surface planar geodesic developable surface 

Mathematics Subject Classification

Primary 53A05 Secondary 53C12 53C22 


  1. 1.
    Bates, L., Melko, O.: On curves of constant torsion. I. J. Geom. 104, 213–227 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bishop, R.: There is more than one way to frame a curve. Am. Math. Mon. 82, 246–251 (1975)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brander, D., Gravesen, J.: Surfaces foliated by planar geodesics: a model for curved wood design. In: Swart, D., Séquin, C., Fenyvesi, K. (eds.) Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture, Phoenix, Arizona, pp. 487–490 (2017). Tessellations Publishing.
  4. 4.
    Chicone, C., Kalton, N.: Flat embeddings of the Möbius strip in\(R^3\). Commun. Appl. Nonlinear Anal. 9, 31–50 (2002)Google Scholar
  5. 5.
    Darboux, G.: Lecons sur la théorie générale des surfaces, vol. I. Gauthier-Villars, Paris (1887)MATHGoogle Scholar
  6. 6.
    Eisenhart, L.: A Treatise on the Differential Geometry of Curves and Surfaces. The Atheneum Press, Boston (1909)MATHGoogle Scholar
  7. 7.
    Fenchel, W.: On the differential geometry of closed space curves. Bull. Am. Math. Soc. 57, 44–54 (1951)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Monge, G.: Application de l’analyse a la géométrie, 5th edn. Bachelier, Paris (1850)Google Scholar
  9. 9.
    Raffy, L.: Sur les surfaces á lignes de courbure planes, dont les plans enveloppent un cylindre. Ann. de l’Éc. Norm. 3(18), 343–370 (1901)MATHGoogle Scholar
  10. 10.
    Randrup, T., Røgen, P.: Sides of the Möbius strip. Arch. Math. 66, 511–521 (1996)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Scherrer, W.: Eine Kennzeichnung der Kugel. Vierteljschr. Naturforsch. Ges. Zürich 85, 40–46 (1940)MathSciNetMATHGoogle Scholar
  12. 12.
    Weiner, J.: Closed curves of constant torsion. II. Proc. Am. Math. Soc. 67, 306–308 (1977)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Wunderlich, W.: Über ein abwickelbares Möbiusband. Monatsh. Math. 66, 276–289 (1962)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkKongens LyngbyDenmark

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