Principal curvature

From Wikipedia, the free encyclopedia

In differential geometry, the two principal curvatures at a given point of a surface measure how the surface bends by different amounts in different directions at that point.

Contents 1 Overview 2 Formal definition 2.1 Generalizations 3 Classification of points on a surface 4 Lines of curvature 5 References 6 External links

[edit] Overview

At each point p of a differentiable surface in 3-dimensional Euclidean space one may choose a unit normal vector. A normal plane at p is one that contains the normal, and will therefore also contain a unique direction tangent to the surface and cut the surface in a plane curve. This curve will in general have different curvatures for different normal planes at p. The principal curvatures at p are the maximum and minimum values of this curvature.

Here the curvature of a curve is taken to be the reciprocal of the radius of the osculating circle. The curvature is taken to be positive if the curve turns in the same direction as the surface's chosen normal, otherwise negative. The directions of the normal plane where the curvature takes its maximum and minimum values are always perpendicular, a result of Euler (1760), and are called principal directions. From a modern perspective, this theorem follows from the spectral theorem because they can be given as the eigenvectors of a symmetric matrix — the second fundamental form of the surface. A systematic analysis of the principal curvatures and principal directions was undertaken by Gaston Darboux, using Darboux frames.

The product k₁k₂ of the two principal curvatures is the Gaussian curvature, K and the average (k₁+k₂)/2 is the mean curvature, H.

If at least one of the principal curvatures is zero at every point, the surface is a developable surface. For a minimal surface, the mean curvature is zero at every point.

[edit] Formal definition

Let M be a surface in Euclidean space with second fundamental form II(X,Y). Fix a point p∈M, and an orthonormal basis X₁, X₂ of tangent vectors at p. Then the principal curvatures are the eigenvalues of the symmetric matrix

If X₁ and X₂ are selected so that the matrix [II_ij] is a diagonal matrix, then they are called the principal directions. If the surface is oriented, then one often requires that the pair (X₁, X₂) to be positively oriented with respect to the given orientation.

[edit] Generalizations

For hypersurfaces in higher dimensional Euclidean spaces, the principal curvatures may be defined in a directly analogous fashion. The principal curvatures are the eigenvalues of the matrix of the second fundamental form II(X_i,X_j) in an orthonormal basis of the tangent space. The principal directions are the corresponding eigenvectors.

Similarly, if M is a hypersurface in a Riemannian manifold N, then the principal curvatures are the eigenvalues of its second-fundamental form. If k₁, ..., k_n are the n principal curvatures at a point p ∈ M and X₁, ..., X_n are corresponding orthonormal eigenvectors (principal directions), then the sectional curvature of M at p is given by

K(X_i,X_j) = k_ik_j.

[edit] Classification of points on a surface

• At elliptical points, both principal curvatures have the same sign, and the surface is locally convex.
  • At umbilic points, both principal curvatures are equal and every tangent vector can be considered a principal direction.
• At hyperbolic points , the principal curvatures have opposite signs, and the surface will be locally saddle shaped.
• At parabolic points, one of the principal curvatures is zero. Parabolic points generally lie in a curve separating elliptical and hyperbolic regions.

[edit] Lines of curvature

The lines of curvature or curvature lines are curves which are always tangent to a principal direction (they are integral curves for the principal curvature line fields). There will be two lines of curvature through each non-umbilic point and the lines will cross at right angles.

In the vicinity of an umbilic the lines of curvature form one of three configurations star, lemon and monstar (derived from lemon-star)^[1]. These points are also called Darbouxian Umbilics, in honor to Gaston Darboux, the first to make a systematic study in Vol. 4, p455, of his Leçons (1896).

configurations of lines of curvature near umbilics

Lemon

Monstar

Star

In these figures, the red curves are the lines of curvature for one family of principal directions, and the blue curves for the other.

[edit] References

• Darboux, Gaston (1887,1889,1896). Leçons sur la théorie génerale des surfaces: Volume I, Volume II, Volume III, Volume IV. Gauthier-Villars. 
• Guggenheimer, Heinrich (1977). "Chapter 10. Surfaces", Differential Geometry. Dover. ISBN 0-486-63433-7. 
• Kobayashi, Shoshichi and Nomizu, Katsumi (1996 (New edition)). Foundations of Differential Geometry, Vol. 2. Wiley-Interscience. ISBN 0471157325. 
• Spivak, Michael (1999). A Comprehensive introduction to differential geometry (Volume 3). Publish or Perish. ISBN 0-914098-72-1. 

1. ^ Berry, M V, & Hannay, J H, 'Umbilic points on Gaussian random surfaces', J.Phys.A 10, 1977, 1809-21, .

[edit] External links

• Historical Comments on Monge's Ellipsoid and the Configuration of Lines of Curvature on Surfaces Immersed in R³

v • d • e Different notions of curvature defined in differential geometry Differential geometry of curves Curvature · Torsion of curves · Frenet-Serret formulas · Radius of curvature (applications) · Affine curvature Differential geometry of surfaces Principal curvatures · Gaussian curvature  · Mean curvature  · Darboux frame  · Gauss–Codazzi equations  · First fundamental form  · Second fundamental form Riemannian geometry Curvature of Riemannian manifolds · Riemann curvature tensor  · Ricci curvature  · Scalar curvature  · Sectional curvature  · Gauss curvature Curvature of connections Curvature form  · Torsion tensor  · Cocurvature  · Holonomy