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Isotopies of generic plane curves

  • J. W. Bruce (a1)
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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].

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Copyright
References
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1.Arnol'd, V. I., Wavefront evolution and equivariant Morse lemma, Comm. Pure Appl. Math. 29 (1976), 557582.
2.Brocker, Th. and Lander, L., Differentiable germs and catastrophes, London Math. Soc. Lecture Note Series, No. 17 (Cambridge University Press, 1975).
3.Bruce, J. W., The duals of generic hypersurfaces, Math. Scand. 49 (1981), 3660.
4.Bruce, J. W., Generic functions on algebraic sets, to appear.
5.Gibson, C. G., Singular points of smooth mappings, Pitman Research Notes in Mathematics, 25 (Pitman, 1979).
6.Porteous, I. R.. The normal singularities of a submanifold, J. Differential Geometry 5 (1971), 543564.
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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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