In this Note we study the envelope of 1-parameter family of smooth curves tangent to a curve having a semicubic cusp, such that the radius of curvature at the tangency point vanishes when this point approaches the cusp. We show that, generically, the closure of the envelope has two semicubic cusps at the same point, one of which is the given cusp, tangent to the same straight line.
Dans cette Note on étudie l'enveloppe d'une famille à un paramètre de courbes lisses tangentes à une courbe ayant un cusp semicubique, telles que le rayon de courbure au point de tangence tende vers zéro lorsque ce point approche le cusp. On montre que, génériquement, l'adhérence de cette enveloppe a deux cusps semicubiques au même point, dont l'un est le cusp donné, tangents à une même droite.
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Gianmarco Capitanio 1
@article{CRMATH_2002__335_3_249_0, author = {Gianmarco Capitanio}, title = {On the envelope of 1-parameter families of curves tangent to a semicubic cusp}, journal = {Comptes Rendus. Math\'ematique}, pages = {249--254}, publisher = {Elsevier}, volume = {335}, number = {3}, year = {2002}, doi = {10.1016/S1631-073X(02)02472-X}, language = {en}, }
Gianmarco Capitanio. On the envelope of 1-parameter families of curves tangent to a semicubic cusp. Comptes Rendus. Mathématique, Volume 335 (2002) no. 3, pp. 249-254. doi : 10.1016/S1631-073X(02)02472-X. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02472-X/
[1] Astroidal geometry of hypocycloids and the Hessian topology of hyperbolic polynomials, Russian Math. Surveys, Volume 56 (2001) no. 6, pp. 1019-1083
[2] Sur la théorie des enveloppes, J. Math. Pures Appl., Volume 41 (1962) no. 9, pp. 177-192
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