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Principal curvature
 
 
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In differential geometry, the two principal curvatures at a given point of a surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
 measure how the surface bends by different amounts in different directions at that point.

At each point p of a differentiable
Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to Euclidean space to allow one to do calculus. This article deals with differentiability in different contexts including: smooth function, k times differentiable, and holomorphic function....
 surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
 in 3-dimensional Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in 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.






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In differential geometry, the two principal curvatures at a given point of a surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
 measure how the surface bends by different amounts in different directions at that point.

At each point p of a differentiable
Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to Euclidean space to allow one to do calculus. This article deals with differentiability in different contexts including: smooth function, k times differentiable, and holomorphic function....
 surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
 in 3-dimensional Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in 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 curvature
Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line , but this is defined in different ways depending on the context....
s 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
Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1⁄x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1....
 of the radius
RADIUS

Remote Authentication Dial In User Service is a networking protocol that provides centralized access, authorization and accounting management for people or computers to connect and use a network service....
 of the osculating circle
Osculating circle

In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point on the curve is the circle whose center lies on the inner normal line and whose curvature is the same as that of the given curve at that point....
. 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
Leonhard Euler

Leonhard Paul Euler was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany.Euler made important discoveries in fields as diverse as calculus and graph theory....
 (1760), and are called principal directions. From a modern perspective, this theorem follows from the spectral theorem
Spectral theorem

In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrix_....
 because they can be given as the eigenvectors of a symmetric matrix
Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix, A, that is equal to its transposeThe entries of a symmetric matrix are symmetric with respect to the main diagonal ....
 — the second fundamental form
Second fundamental form

In differential geometry, the second fundamental form is a quadratic form on the tangent plane of a differential geometry of surfaces in the three dimensional Euclidean space, usually denoted by II....
 of the surface. A systematic analysis of the principal curvatures and principal directions was undertaken by Gaston Darboux, using Darboux frame
Darboux frame

In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet-Serret formulas as applied to surface geometry....
s.

The product k1k2 of the two principal curvatures is the Gaussian curvature
Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, ?1 and ?2, of the given point....
, K, and the average (k1+k2)/2 is the mean curvature
Mean curvature

In mathematics, the mean curvature of a surface is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedding surface in some ambient space such as Euclidean space....
, H.

If at least one of the principal curvatures is zero at every point, the surface is a developable surface
Developable surface

In mathematics, a developable surface is a surface with zero Gaussian curvature. That is, it is "surface" that can be Flatness onto a Plane without distortion ....
. For a minimal surface
Minimal surface

In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....
, the mean curvature is zero at every point.

Formal definition

Let M be a surface in Euclidean space with second fundamental form
Second fundamental form

In differential geometry, the second fundamental form is a quadratic form on the tangent plane of a differential geometry of surfaces in the three dimensional Euclidean space, usually denoted by II....
 II(X,Y). Fix a point p∈M, and an orthonormal basis
Orthonormal basis

In mathematics, an orthonormal basis of an inner product space V , is a set of mutually orthogonality vectors of magnitude 1 that span the space when infinite linear combinations are allowed....
 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 [IIij] is a diagonal matrix, then they are called the principal directions. If the surface is oriented
Orientation (mathematics)

In mathematics, an orientation on a real number vector space is a choice of which ordered basis are "positively" oriented and which are "negatively" 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(Xi,Xj) 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
Riemannian manifold

In Riemannian geometry, a Riemannian manifold is a real differentiable manifold M in which each tangent space is equipped with an Inner product space g in a manner which varies smoothly from point to point....
 N, then the principal curvatures are the eigenvalues of its second-fundamental form. If k1, ..., kn are the n principal curvatures at a point p ∈ M and X1, ..., Xn are corresponding orthonormal eigenvectors (principal directions), then the sectional curvature
Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature depends on a two-dimensional plane in the tangent space at p....
 of M at p is given by

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
      Monkey saddle

      In mathematics, the monkey saddle is the surface defined by the equationIt belongs to the class of saddle surfaces and its name derives from the observation that a saddle for a monkey requires three depressions: two for the legs, and one for the tail....
       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 curve
Integral curve

In mathematics, an integral curve for a vector field defined on a manifold is a curve in the manifold whose tangent vector at each point along the curve is the vector field itself at that point....
s 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). 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).

Image:TensorLemon.png|Lemon Image:TensorMonstar.png|Monstar Image:TensorStar.png|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
Ridge (differential geometry)

For a smooth surface in three dimensions a ridge point occurs when a Principal curvature#line of curvature has a local maximum or minimum of principal curvature....
. 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.

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