In the mathematical field of differential geometry, 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).
More precisely, 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 (unit) 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, then
Eisenhart, Luther P. (2004), A Treatise on the Differential Geometry of Curves and Surfaces, Dover, ISBN0-486-43820-1 Full 1909 text (now out of copyright)
Euler, Leonhard (1760), "Recherches sur la courbure des surfaces", Mémoires de l'Académie des Sciences de Berlin (published 1767), 16: 119–143.
Spivak, Michael (1999), A comprehensive introduction to differential geometry, Volume II, Publish or Perish Press, ISBN0-914098-71-3
euler, theorem, differential, geometry, mathematical, field, differential, geometry, euler, theorem, result, curvature, curves, surface, theorem, establishes, existence, principal, curvatures, associated, principal, directions, which, give, directions, which, . In the mathematical field of differential geometry 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 More precisely 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 unit 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 kX when regarded as a curve inside PX Provided not all kX are equal there is some unit vector X1 for which k1 kX1 is as large as possible and another unit vector X2 for which k2 kX2 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 8 with X1 then k X k 1 cos 2 8 k 2 sin 2 8 displaystyle kappa X k 1 cos 2 theta k 2 sin 2 theta 1 The quantities k1 and k2 are called the principal curvatures and X1 and X2 are the corresponding principal directions Equation 1 is sometimes called Euler s equation Eisenhart 2004 p 124 See also EditDifferential geometry of surfaces Dupin indicatrixReferences EditEisenhart Luther P 2004 A Treatise on the Differential Geometry of Curves and Surfaces Dover ISBN 0 486 43820 1 Full 1909 text now out of copyright Euler Leonhard 1760 Recherches sur la courbure des surfaces Memoires de l Academie des Sciences de Berlin published 1767 16 119 143 Spivak Michael 1999 A comprehensive introduction to differential geometry Volume II Publish or Perish Press ISBN 0 914098 71 3 nbsp This differential geometry related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Euler 27s theorem differential geometry amp oldid 1051501933, wikipedia, wiki, book, books, library,
