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).
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.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.
Bruce, J.W. and Giblin, P.J. 1986. Growth, motion and 1-parameter families of symmetry sets. Proc. Royal Soc. Edinburgh, 104A:179–204.
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.
Author information
Authors and Affiliations
Additional information
The third author was supported by the Esprit grant VIVA while this paper was in preparation.
Rights and permissions
About this article
Cite this article
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
Issue Date:
DOI: https://doi.org/10.1007/BF00128235