# Théorème d'Euler (géométrie différentielle)

Théorème d'Euler (géométrie différentielle) Dans le domaine mathématique de la géométrie différentielle, Euler's theorem is a result on the curvature of curves on a surface. The theorem establishes the existence of principal curvatures and associated principal directions which give the directions in which the surface curves the most and the least. The theorem is named for Leonhard Euler who proved the theorem in (Euler 1760).

Plus précisément, let M be a surface in three-dimensional Euclidean space, and p a point on M. A normal plane through p is a plane passing through the point p containing the normal vector to M. Through each (unité) tangent vector to M at p, there passes a normal plane PX which cuts out a curve in M. That curve has a certain curvature κX when regarded as a curve inside PX. Provided not all κX are equal, there is some unit vector X1 for which k1 = κX1 is as large as possible, and another unit vector X2 for which k2 = κX2 is as small as possible. Euler's theorem asserts that X1 and X2 are perpendicular and that, moreover, if X is any vector making an angle θ with X1, alors {displaystyle kappa _{X}=k_{1}car ^{2}theta +k_{2}péché ^{2}thêta .,} (1) The quantities k1 and k2 are called the principal curvatures, and X1 and X2 are the corresponding principal directions. Équation (1) is sometimes called Euler's equation (Eisenhart 2004, p. 124).

See also Differential geometry of surfaces Dupin indicatrix References Eisenhart, Luther P. (2004), A Treatise on the Differential Geometry of Curves and Surfaces, Douvres, ISBN 0-486-43820-1 Full 1909 texte (now out of copyright) Euler, Leonhard (1760), "Recherches sur la courbure des surfaces", Mémoires de l'Académie des Sciences de Berlin (publié 1767), 16: 119–143. Spivak, Michael (1999), Une introduction complète à la géométrie différentielle, Tome II, Publier ou périr Presse, ISBN 0-914098-71-3 Cet article lié à la géométrie différentielle est un bout. Vous pouvez aider Wikipédia en l'agrandissant.

Catégories: Differential geometry of surfacesTheorems in differential geometryLeonhard EulerDifferential geometry stubs

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