On the envelope of 1-parameter families of curves tangent to a semicubic cusp

Gianmarco Capitanio

doi:10.1016/S1631-073X(02)02472-X

On the envelope of 1-parameter families of curves tangent to a semicubic cusp
[Sur l'enveloppe des familles à un paramètre de courbes tangentes à un cusp semicubique]

Gianmarco Capitanio ¹

¹ Équipe de géométrie et dynamique, Université Paris VII, 175, rue du Chevaleret, 75013 Paris, France

Comptes Rendus. Mathématique, Volume 335 (2002) no. 3, pp. 249-254.

Abstract (VO)
Résumé

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.

Reçu le : 2002-05-16
Accepté le : 2002-06-03
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02472-X

Affiliations des auteurs :

Gianmarco Capitanio ¹

¹ Équipe de géométrie et dynamique, Université Paris VII, 175, rue du Chevaleret, 75013 Paris, France

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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/

Bibliographie
Cité par

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Kelly Bickel; Pamela Gorkin; Trung Tran Applications of envelopes, Complex Analysis and its Synergies, Volume 6 (2020) no. 1, p. 14 (Id/No 2) | DOI:10.1007/s40627-019-0039-z | Zbl:1465.53018
Thierry Dana-Picard; Nurit Zehavi Automated study of envelopes of one-parameter families of surfaces, Applications of computer algebra, Kalamata, Greece, July 20–23, 2015, Cham: Springer, 2017, pp. 29-44 | DOI:10.1007/978-3-319-56932-1_4 | Zbl:1386.65088
Masatomo Takahashi Envelopes of Legendre curves in the unit tangent bundle over the Euclidean plane, Results in Mathematics, Volume 71 (2017) no. 3-4, pp. 1473-1489 | DOI:10.1007/s00025-016-0619-7 | Zbl:1378.58034
Thierry Dana-Picard; Nurit Zehavi Revival of a classical topic in differential geometry: the exploration of envelopes in a computerized environment, International Journal of Mathematical Education in Science and Technology, Volume 47 (2016) no. 6, pp. 938-959 | DOI:10.1080/0020739x.2015.1133852 | Zbl:1345.97009
Gianmarco Capitanio Stable tangential family germs and singularities of their envelopes, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 341 (2005) no. 8, pp. 503-508 | DOI:10.1016/j.crma.2005.09.014 | Zbl:1088.53001
G. Capitanio Singularities of the envelope of curves tangent to a semi-cubic cusp, Journal of Mathematical Sciences, Volume 126 (2005) no. 4, p. 1243 | DOI:10.1007/pl00021949
G. Capitanio Singularities of the envelope of curves tangent to a semi-cubic cusp, Journal of Mathematical Sciences (New York), Volume 126 (2005) no. 4, pp. 1243-1250 | DOI:10.1007/s10958-005-0074-6 | Zbl:1079.53004
Saralees Nadarajah; Dipak K. Dey Multitude of multivariate $t$ -distributions, Statistics, Volume 39 (2005) no. 2, pp. 149-181 | DOI:10.1080/02331880500031407 | Zbl:1067.62059

Cité par 8 documents. Sources : Crossref, zbMATH

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