Skip to main content Accessibility help
×
×
Home

Wavefronts and parallels in Euclidean space

  • J. W. Bruce
Extract

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).

Copyright
References
Hide All
(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).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed