Journal of the Mathematical Society of Japan

Contact of a regular surface in Euclidean 3-space with cylinders and cubic binary differential equations

Toshizumi FUKUI, Masaru HASEGAWA, and Kouichi NAKAGAWA

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We investigate the contact types of a regular surface in the Euclidean 3-space $\mathbb{R}^3$ with right circular cylinders. We present the conditions for existence of cylinders with $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $D_4$, and $D_5$ contacts with a given surface. We also investigate the kernel field of $A_{\ge 3}$-contact cylinders on the surface. This is defined by a cubic binary differential equation and we classify singularity types of its flow in the generic context.

Article information

J. Math. Soc. Japan, Volume 69, Number 2 (2017), 819-847.

First available in Project Euclid: 20 April 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A05: Surfaces in Euclidean space 53A60: Geometry of webs [See also 14C21, 20N05] 58K05: Critical points of functions and mappings 58C27

contact with cylinders Monge cubic cylindrical directions


FUKUI, Toshizumi; HASEGAWA, Masaru; NAKAGAWA, Kouichi. Contact of a regular surface in Euclidean 3-space with cylinders and cubic binary differential equations. J. Math. Soc. Japan 69 (2017), no. 2, 819--847. doi:10.2969/jmsj/06920819.

Export citation


  • T. Banchoff, T. Gaffney and C. McCrory, Cusps of Gauss mappings, Res. Notes Math., 55, Pitman, 1982.
  • J. W. Bruce and P. J. Giblin, Curves and Singularities, Cambridge University Press, 1984, Second Edition, 1992.
  • J. W. Bruce, P. J. Giblin and F. Tari, Families of surfaces: height functions, Gauss maps and duals, In: Real and Complex Singularities, (ed. W. L. Marar), Pitman Res. Notes Math. Ser., 333, Longman, Harlow, 1995.
  • J. W. Bruce and F. Tari, Dupin indicatrices and families of curve congruences, Trans. Amer. Math. Soc., 357 (2005), 267–285.
  • G. Darboux, Leçons sur la théorie générale des surfaces, IV, Gauthier-Villars, Paris, 1896.
  • S. Izumiya, Differential geometry from the viewpoint of Lagrangian or Legendrian singularity theory, In: Singularity theory, Proceedings of the Singularity School and Conference, Marceille, 2005 (eds. D. Chéniot, N. Dutertre, C. Murolo, D. Trotman and A. Pichon), World Scientific Pub. Co. Inc., 2007.
  • S. Izumiya, M. C. Romero Fuster, M. A. Ruas and F. Tari, Differential geometry from singularity theory viewpoint, World Scientific Pub. Co. Inc., 2015.
  • D. K. H. Mochida, M. C. Romero Fuster and M. A. Ruas, The geometry of surfaces in 4-space from a contact viewpoint, Geometriae Dedicata, 54 (1995), 323–332.
  • R. A. Garcia, D. K. H. Mochida, M. C. Romero Fuster and M. A. Ruas, Inflection points and topology of surfaces in 4-space, Trans. Amer. Math. Soc., 352 (2000), 3029–3043.
  • K. Kakié, The resultant of several homogeneous polynomials in two indeterminates, Proc. Amer. Math. Soc., 54 (1976), 1–7.
  • J. J. Koenderink, What does the occluding contour tell us about solid shape?, Perception, 13 (1984), 321–330.
  • J. J. Koenderink, Solid shape, MIT Press Series in Artificial Intelligence, MIT Press, Cambridge, MA, 1990.
  • J. Mather, Stability of mappings, III, Finitely determined map-germs, Publ. Math. I.H.E.S., 35 (1969), 127–156.
  • J. Montaldi, On contact between submanifolds, Michigan Math. J., 33 (1986), 195–199.
  • I. R. Porteous, The normal singularities of a submanifold, J. Differential Geometry, 5 (1971), 543–564.
  • I. R. Porteous, The normal singularities of surfaces in $\mathbb{R}^3$, Singularities, Part 2 (Arcata, Calif., 1981), 379–393, Proc. Sympos. Pure Math., 40, Amer. Math. Soc., Providence, RI, 1983.
  • T. Takahashi, Homogeneous hypersurfaces in space of constant curvature, J. Math. Soc. Japan, 22 (1970), 395–410.
  • R. Thom, Stabilité structurelle et morphogenèse, Benjamin Paris, 1972.