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Wavefronts and parallels in Euclidean space

  • J. W. Bruce

Given a smooth plane curve or surface in ℝ3 its parallels consist of those curves or surfaces a fixed distance d down the normals in a fixed direction. Generically they have Legendre singularities. We are concerned here with the way in which these parallels change as we alter the distance d. (Alternatively the manner in which wave-fronts change as they evolve from an initial smooth wavefront.)

This problem was considered in (1) by V. I. Arnold. In a very beautiful paper he describes the generic evolution of wavefronts but does not prove that for a generic initial wavefront in ℝ2 or ℝ3 the evolution is of the type described there. This we do here, using the tool of transversality. A more positive outcome of our investigation is that some of Arnold's generic forms do not occur (those corresponding to A2 singularities).

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(1)Arnold, V. I.Wavefront evolution and equivariant Morse lemma. Comm. Pure Appl. Math. 29 (1976), 557582.
(2)Bruce, J. W., Giblin, P. J. and Gibson, C. G.Source genericity of caustics by reflexion in ℝ3. Proc. Royal Soc. London A 381, 83116 (1982).
(3)Gibson, C. G.Singular points of smooth mappings. Pitman Research Notes in Mathematics, no. 25 (1979).
(4)Looijenga, E. J. N. Structural stability of families of C -functions. (Thesis, University of Amsterdam, 1974.)
(5)Porteous, I. R.The normal singularities of submanifold. J. Differential Geom. 5 (1971), 543564.
(6)Wall, C. T. C. Geometric properties of generic differentiable manifolds. In Geometry and Topology, vol. III, Springer Lecture Notes in Mathematics, no. 597 (1976).
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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