Geodesic
Isotopies of generic plane curves
Glasgow mathematical journal, Tome 24 (1983) no. 2, pp. 195-206

Voir la notice de l'article provenant de la source Cambridge University Press

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]. 
Bruce, J. W. Isotopies of generic plane curves. Glasgow mathematical journal, Tome 24 (1983) no. 2, pp. 195-206. doi: 10.1017/S0017089500005292
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[1] 1.Arnol'd, V. I., Wavefront evolution and equivariant Morse lemma, Comm. Pure Appl. Math. 29 (1976), 557–582. Google Scholar | DOI

[2] 2.Brocker, Th. and Lander, L., Differentiable germs and catastrophes, London Math. Soc. Lecture Note Series, No. 17 (Cambridge University Press, 1975). Google Scholar | DOI

[3] 3.Bruce, J. W., The duals of generic hypersurfaces, Math. Scand. 49 (1981), 36–60. Google Scholar | DOI

[4] 4.Bruce, J. W., Generic functions on algebraic sets, to appear. Google Scholar

[5] 5.Gibson, C. G., Singular points of smooth mappings, Pitman Research Notes in Mathematics, 25 (Pitman, 1979). Google Scholar

[6] 6.Porteous, I. R.. The normal singularities of a submanifold, J. Differential Geometry 5 (1971), 543–564. Google Scholar | DOI

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