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Wavefronts and parallels in Euclidean space

Published online by Cambridge University Press:  24 October 2008

J. W. Bruce
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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

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References

REFERENCES

(1)Arnold, V. I.Wavefront evolution and equivariant Morse lemma. Comm. Pure Appl. Math. 29 (1976), 557–582.CrossRefGoogle Scholar
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