Smooth Surfaces and Their Projections

  • Vladimir Igorevich Arnold


A smooth curve on the plane can have a tangent which touches it tangentially at any number of points (Fig. 56), but this is not true for generic curves. By a small perturbation of the curve one can always achieve that no straight line will be tangential at more than two points.


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Bibliographical Comments

  1. Y. L. Kergosien, R. Thorn: Sur les points paraboliques des surfaces. C. R. Acad. Sel., Paris, Ser. A 290 (1980), 705–710.zbMATHGoogle Scholar
  2. Y. L. Kergosien: La famille des projections orthogonales d’une surface et ses singularités. C. R. Acad. Sel., Paris, Ser. I 292 (1981), 929–932.)MathSciNetzbMATHGoogle Scholar
  3. O. A. Platonova: Singularities of projections of smooth surfaces. Usp. Mat. Nauk 39:1 (1984), 149–150 (English translation: Russ. Math. Surv. 39:1 (1984), 177-178).MathSciNetzbMATHGoogle Scholar
  4. O. A. Platonova: Singularities of the mutual disposition of a surface and a line. Usp. Mat. Nauk 36:1 (1981), 221–222 (English translation: Russ. Math. Surv. 36:1 (1981), 248-249).MathSciNetzbMATHGoogle Scholar
  5. O. A. Platonova: Projections of smooth surfaces (in Russian). Tr. Semin. Im. I. G. Petrovskogo 10 (1984), 135–149.MathSciNetGoogle Scholar
  6. O. A. Platonova’s Thesis (in Russian). Moscow State University, 1981, 150 p.).Google Scholar
  7. E. E. Landis: Tangential singularities. Funkts. Anal. Prilozh. 15:2 (1981), 36–49 (English translation: Funct. Anal. Appl. 15 (1981), 103-114).MathSciNetGoogle Scholar
  8. E. E. Landis’s Thesis (in Russian). Moscow State University, 1983, 142 p.).Google Scholar
  9. V. I. Arnol’d: Singularities of systems of rays. Usp. Mat. Nauk 38:2 (1983), 77–147 (English translation: Russ. Math. Surv. 38:2 (1983), 87-176).Google Scholar
  10. O. P. Shcherbak: Projectively dual space curves and Legendre singularities (in Russian). Tr. Tbilis. Univ. 232-233, ser. Mat. Mekh. Astron. 13-14 (1982), 280–336.Google Scholar

The proofs of the theorems on the projections depend on :

  1. V. I. Arnol’d: Indices of singular points of 1-forms on a manifold with boundary, convolution of invariants of reflection groups, and singular projections of smooth surfaces. Usp. Mat. Nauk 34:2 (1979), 3–38 (English translation: Russ. Math. Surv. 34:2 (1979), 1-42).Google Scholar

Other approaches:

  1. T. Banchoff, T. Gaffney, C. McCrory: Cusps of Gauss mappings. Res. Notes Math. 55, Pitman, Boston-London-Melbourne 1982.Google Scholar

A survey on singularities of projections:

  1. V. V. Goryunov: Singularities of projections of full intersections, in: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. (Contemporary Problems of Mathematics) 22, Viniti, Moscow 1983, pp. 167–206 (English translation: J. Sov. Math. 27 (1984), 2785-2811).MathSciNetGoogle Scholar

See also:

  1. V. V. Goryunov: Geometry of bifurcation diagrams of simple projections onto the line. Funkts. Anal. Prilozh. 15:2 (1981), 1–8 (English translation: Funct. Anal. Appl. 15 (1981), 77-82).MathSciNetGoogle Scholar
  2. V. V. Goryunov: Projection of 0-dimensional complete intersections onto a line and the k(π,1) conjecture. Usp. Mat. Nauk 37:3 (1982), 179–180 (English translation: Russ. Math. Surv. 37:3 (1982), 206-208).MathSciNetGoogle Scholar
  3. V. V. Goryunov: Bifurcation diagrams of some simple and quasihomogeneous singularities. Funkts. Anal. Prilozh. 17:2 (1983), 23–37 (English translation: Funct. Anal. Appl. 17 (1983), 97-108).MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Vladimir Igorevich Arnold
    • 1
  1. 1.Department of MathematicsUniversity of MoscowMoscowUSSR

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