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Parabolic curves of evolving surfaces

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Abstract

In this article we show how certain geometric structures which are also associated with a smooth surface evolve as the shape of the surface changes in a 1-parameter family. We concentrate on the parabolic set and its image under the Gauss map, but the same techniques also classify the changes in the dual of the surface. All these have significance for computer vision, for example through their connection with specularities and apparent contours. With the aid of our complete classification, which includes all the phenomena associated with multi-contact tangent planes as well as those associated with parabolic sets, we re-examine examples given by J. Koenderink in his book (1990) under the title of Morphological Scripts.

We also explain some of the connections between parabolic sets and ‘ridges’ of a surface, where principal curvatures achieve turning values along lines of curvature.

The point of view taken is the analysis of the contact between surfaces and their tangent planes. A systematic investigation of this yields the results using singularity theory. The mathematical details are suppressed here and appear in Bruce et al. (1993).

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References

  • Arnold, V.I., Varchenko, A.N., and Gusein-Zade, S.M. 1985. Singularities of Differentiable Mappings, Vol. I. Birkhäuser.

  • Banchoff, T., Gaffney, T., and McCrory, C. 1982. Cusps of Gauss Mappings. Pitman.

  • Bruce, J.W. 1994. Generic geometry, transversality and projections. J. London Math. Soc., 49:183–194.

    Google Scholar 

  • Bruce, J.W. and Giblin, P.J. 1986. Growth, motion and 1-parameter families of symmetry sets. Proc. Royal Soc. Edinburgh, 104A:179–204.

    Google Scholar 

  • Bruce, J.W. and Giblin, P.J. 1992. Curves and Singularities. Cambridge University Press, 2nd edition.

  • Bruce, J.W., Giblin, P.J., and Tari, F. 1995. Families of surfaces: height functions, Gauss maps and duals, in Real and Complex Singularities, ed. W.L. Marar, Pitman Research Notes in Mathematics 333, pp. 148–178.

  • Hilbert, D. and Cohn-Vossen, S. 1952. Geometry and the Imagination. Chelsea Publications.

  • Koenderink, J.J. 1990. Solid Shape. MIT Press.

  • Porteous, I.R. 1994. Geometric Differentiation. Cambridge University Press, (in press).

  • Rieger, J.H. 1993. Projections of generic surfaces of revolution. Geom. Dedicata, 48:211–230.

    Google Scholar 

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Authors and Affiliations

  1. Department of Pure Mathematics, The University of Liverpool, P.O. Box 147, L69 3BX, Liverpool, England

    J. W. Bruce, P. J. Giblin & F. Tari

Authors
  1. J. W. Bruce
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  2. P. J. Giblin
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  3. F. Tari
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Additional information

The third author was supported by the Esprit grant VIVA while this paper was in preparation.

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Bruce, J.W., Giblin, P.J. & Tari, F. Parabolic curves of evolving surfaces. Int J Comput Vision 17, 291–306 (1996). https://doi.org/10.1007/BF00128235

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  • DOI: https://doi.org/10.1007/BF00128235

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