Vertices and Inflexions of Plane Sections of Surfaces in ℝ3

  • André Diatta
  • Peter Giblin
Conference paper
Part of the Trends in Mathematics book series (TM)


We discuss the behavior of vertices and inflexions of one-parameter families of plane curves which include a singular member. These arise as sections of smooth surfaces by families of planes parallel to the tangent plane at a given point. We cover all the generic cases, namely elliptic, umbilic, hyperbolic, parabolic and cusp of Gauss points. This work is preliminary to an investigation of symmetry sets and medial axes for these families of curves, reported elsewhere.


Isophote curve symmetry set medial axis skeleton vertex inflexion plane curve shape analysis 


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  1. [1]
    T.F. Banchoff, T. Gaffney and C. McCrory, Cusps of Gauss Mappings, Pitman Research Notes in Mathematics, 55, 1982.Google Scholar
  2. [2]
    J.W. Bruce and P.J. Giblin, Curves and Singularities Cambridge University Press, 2nd ed. (1992).Google Scholar
  3. [3]
    J.W. Bruce and P.J. Giblin, ‘Growth, motion and one-parameter families of symmetry sets’, Proc. Royal Soc.Edinburgh 104A (1986), 179–204.MathSciNetGoogle Scholar
  4. [4]
    J.W. Bruce, P.J. Giblin and F. Tari, ‘Parabolic curves of evolving surfaces,’ Int. J. Computer Vision. 17 (1996), 291–306.CrossRefGoogle Scholar
  5. [5]
    J.W. Bruce, P.J. Giblin and F. Tari, ‘Families of surfaces: height functions, Gauss maps and duals’, in Real and Complex Singularities, W.L. Marar (ed.), Pitman Research Notes in Mathematics, Vol. 333 (1995), 148–178.Google Scholar
  6. [6]
    A. Diatta, P.J. Giblin, ‘Geometry of isophote curves’, Scale Space Theory and PDE Methods in Computer Vision 2005. Lecture Notes in Computer Science 3459 (2005), 50–61.Google Scholar
  7. [7]
    A. Diatta, P.J. Giblin, B. Guilfoyle and W. Klingenberg, ‘Plane sections of surfaces and applications to symmetry sets’, Mathematics of Surfaces XI, Lecture Notes in Computer Science 3604 (2005), 147–160.Google Scholar
  8. [8]
    M.D. Garay, ‘On vanishing inflection points of plane curves’, Ann. Inst. Fourier, Grenoble 52 (2002), 849–880.MathSciNetGoogle Scholar
  9. [9]
    P.J. Giblin, ‘Symmetry sets and medial axes in two and three dimensions’, The Mathematics of Surfaces IX, Roberto Cipolla and Ralph Martin (eds.), Springer-Verlag 2000, pp. 306–321.Google Scholar
  10. [10]
    P.L. Hallinan, G.G. Gordon, A.L. Yuille, P. Giblin and D. Mumford, Two and three dimensional patterns of the face, viii+262 pages, Natick, Massachusetts: A.K. Peters 1999.zbMATHGoogle Scholar
  11. [11]
    J.J. Koenderink, Solid Shape, M.I.T. Press (1990)Google Scholar
  12. [12]
    R.J. Morris, Liverpool Surface Modelling Package, also known as SingSurf, See also R.J. Morris, ‘The use of computer graphics for solving problems in singularity theory’, in Visualization in Mathematics, H.-C.Hege & K.Polthier, Heidelberg: Springer-Verlag (1997), 173–187.Google Scholar
  13. [13]
    I.R. Porteous, Geometric Differentiation, Cambridge University Press, 1994 and 2001.Google Scholar
  14. [14]
    R. Uribe-Vargas, “On the stability of bifurcation diagrams of vanishing flattening points”, Functional Analysis and its Applications 37 (2003), 236–240.CrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • André Diatta
    • 1
  • Peter Giblin
    • 1
  1. 1.University of LiverpoolLiverpoolEngland

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