Principal curvature

Saddle surface with normal planes in directions of principal curvatures

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

Discussion

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, denoted k₁ and k₂, are the maximum and minimum values of this curvature.

Here the curvature of a curve is by definition 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, and 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 principal axes of a symmetric tensor—the second fundamental form. 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, then the Gaussian curvature will be 0 and the surface is a developable surface. For a minimal surface, the mean curvature is zero at every point.

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

$\left[I\!I_{ij}\right] = \begin{bmatrix} I\!I(X_1,X_1)&I\!I(X_1,X_2)\\ I\!I(X_2,X_1)&I\!I(X_2,X_2) \end{bmatrix}.$

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.

Without reference to a particular orthonormal basis, the principal curvatures are the eigenvalues of the shape operator, and the principal directions are its eigenvectors.

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.

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. These typically occur in isolated points.
• 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.
  • At flat umbilic points both principal curvatures are zero. A generic surface will not contain flat umbilic points. The Monkey saddle is one surface with an isolated flat umbilic.

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 direction 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.

When a line of curvature has a local extremum of the same principal curvature then the curve has a ridge point. These ridge points form curves on the surface called ridges. The ridge curves pass through the umbilics. For the star pattern either 3 or 1 ridge line pass through the umbilic, for the monstar and lemon only one ridge passes through.^[2]

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, .
2. ^ Porteous, I. R. (1994). Geometric Differentiation. Cambridge University Press. ISBN 0-521-39063-X 

External links

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