Abstract
We prove that the level sets of a real C s function of two variables near a non-degenerate critical point are of class C [s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at an elliptic or hyperbolic point, and in particular at an umbilic point. We go on to use the results to study symmetry sets of the planar sections. We also analyse one of the cases coming from a degenerate critical point, corresponding to an elliptic cusp of Gauss on a surface, where the differentiability is reduced to C [s/4]. However in all our applications we assume C ∞ smoothness.
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Banchoff, T.F., Gaffney, T., McCrory, C.: Cusps of Gauss Mappings. Pitman Research Notes in Mathematics, vol. 55 (1982)
Bruce, J.W., Giblin, P.J.: Growth, motion and one-parameter families of symmetry sets. In: Proc. Royal Soc. Edinburgh, vol. 104A, pp. 179–204 (1986)
Bruce, J.W., Giblin, P.J., Tari, F.: Ridges, crests and sub-parabolic lines of evolving surfaces. Int. J. Computer Vision 18, 195–210 (1996)
Diatta, A., Giblin, P.J.: Vertices and inflexions of plane sections of surfaces in ℝ3’. In: Real and Complex Singularities 2004. LMS Lecture Notes in Maths, Cambridge University Press, Cambridge (2004) (to appear), http://www.liv.ac.uk/~pjgiblin
Diatta, A., Giblin, P.J.: Geometry of isophote curves. In: Kimmel, R., Sochen, N.A., Weickert, J. (eds.) Scale-Space 2005. LNCS, vol. 3459, pp. 50–61. Springer, Heidelberg (2005), Available from http://www.liv.ac.uk/~pjgiblin
Giblin, P.J.: Symmetry sets and medial axes in two and three dimensions. In: Cipolla, R., Martin, R. (eds.) The Mathematics of Surfaces IX, pp. 306–321. Springer, Heidelberg (2000)
Koenderink, J.J.: Solid Shape. MIT Press, Cambridge (1990)
Morris, R.J.: Liverpool Surface Modelling Package, http://www.amsta.leeds.ac.uk/~rjm/lsmp/ ; See also Morris, R.J.: The use of computer graphics for solving problems in singularity theory. In: Hege, H.-C., Polthier, K. (eds.) Visualization in Mathematics, pp. 173–187. Springer, Heidelberg (1997), http://www.scs.leeds.ac.uk/pfaf/lsmp/SingSurf.html
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Diatta, A., Giblin, P., Guilfoyle, B., Klingenberg, W. (2005). Level Sets of Functions and Symmetry Sets of Surface Sections. In: Martin, R., Bez, H., Sabin, M. (eds) Mathematics of Surfaces XI. Lecture Notes in Computer Science, vol 3604. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11537908_9
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DOI: https://doi.org/10.1007/11537908_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28225-9
Online ISBN: 978-3-540-31835-4
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