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Isotopies of generic plane curves

Published online by Cambridge University Press:  18 May 2009

J. W. Bruce
J. W. Bruce
Affiliation:
University College, Cork, Ireland.
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The aim of this paper is to explore some facets of the geometry of generic isotopies of plane curves. Our major tool will be the paper of Arnol'd [1] on the evolution of wavefronts. The sort of questions one can ask are: in a generic isotopy of a plane curve how are vertices created and destroyed? How does the dual evolve? How can the Gauss map change? In attempting to answer these questions we are taking advantage of the fact that these phenomena are all naturally associated with singularities of type Ak. Now the bifurcation set of an Ak+1 singularity and the discriminant set of an Ak singularity coincide. So we can apply Arnol'd's results on one parameter families of Legendre (discriminant) singularities (e.g. the duals) to get information on one parameter families of Lagrange (bifurcation) singularities (e.g. the evolutes). For bifurcation sets of functions with singularities other than those of type Ak one runs up against problems with smooth moduli—see [4].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 24 , Issue 2 , July 1983 , pp. 195 - 206
Copyright
Copyright © Glasgow Mathematical Journal Trust 1983

References

REFERENCES

1.Arnol'd, V. I., Wavefront evolution and equivariant Morse lemma, Comm. Pure Appl. Math. 29 (1976), 557–582.CrossRefGoogle Scholar
2.Brocker, Th. and Lander, L., Differentiable germs and catastrophes, London Math. Soc. Lecture Note Series, No. 17 (Cambridge University Press, 1975).CrossRefGoogle Scholar
3.Bruce, J. W., The duals of generic hypersurfaces, Math. Scand. 49 (1981), 36–60.CrossRefGoogle Scholar
4.Bruce, J. W., Generic functions on algebraic sets, to appear.Google Scholar
5.Gibson, C. G., Singular points of smooth mappings, Pitman Research Notes in Mathematics, 25 (Pitman, 1979).Google Scholar
6.Porteous, I. R.. The normal singularities of a submanifold, J. Differential Geometry 5 (1971), 543–564.CrossRefGoogle Scholar