Surface de Catalan

Surface de Catalan.

Une surface de Catalan est une surface réglée dont les génératrices restent parallèles à un plan fixe appelé plan directeur. Ces surfaces ont été étudiées par le mathématicien belge Eugène Charles Catalan (1814-1894).

Les équations paramétriques d'une surface de Catalan de plan directeur sont données par[1]

  • avec
.

Chaque droite avec le paramètre fixé, est une génératrice, décrit la courbe directrice et les vecteurs sont tous parallèles au plan directeur.

Si est différentiable, on peut exprimer la condition de parallélisme au plan par

.

Exemples

(1) Les plans :
La courbe directrice est une droite.
(2) Les cylindres:
La courbe directrice est un cercle. Comme plan directeur, on peut prendre tout plan parallèle à l'axe des z.
(3) L'hélicoïde:
La courbe directrice est une hélice et le plan directeur est parallèle au plan x-y.
Cette surface peut aussi être définie à partir d'une droite (axe des z) comme courbe directrice:
(4) Si toutes les génératrices d'une surface de Catalan coupent une droite fixe, alors la surface est appelée un conoïde.

Catalan a prouvé que l'hélicoïde et le plan sont les seules surfaces minimales réglées.

Références

Bibliographie

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