Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T14:16:59.241Z Has data issue: false hasContentIssue false
  • English
  • Français

Wavefronts and parallels in Euclidean space

Published online by Cambridge University Press:  24 October 2008

J. W. Bruce
Get access Rights & Permissions [Opens in a new window]

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).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 93 , Issue 2 , March 1983 , pp. 323 - 333
Copyright
Copyright © Cambridge Philosophical Society 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Arnold, V. I.Wavefront evolution and equivariant Morse lemma. Comm. Pure Appl. Math. 29 (1976), 557–582.CrossRefGoogle Scholar
(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, 83–116 (1982).Google Scholar
(3)Gibson, C. G.Singular points of smooth mappings. Pitman Research Notes in Mathematics, no. 25 (1979).Google Scholar
(4)Looijenga, E. J. N. Structural stability of families of C ∞-functions. (Thesis, University of Amsterdam, 1974.)Google Scholar
(5)Porteous, I. R.The normal singularities of submanifold. J. Differential Geom. 5 (1971), 543–564.CrossRefGoogle Scholar
(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).Google Scholar