Principal curvature

Principal curvature is a term used in differential geometry. Each point of a surface in three-dimensional Euclidean space can be assigned to two principal curvatures.

Definition

Given a point of a regular surface in. Each tangential direction, so any direction that can take at this point is a tangent vector is assigned to the normal curvature: It refers to the curvature of the plane curve that results from a normal section, ie by a section of the given surface with and through the surface normal vector the given tangential certain level. The minimum value and the maximum value of this curvature is referred to as the two principal curvatures and. The associated tangential directions are called principal directions.

Examples

  • For a sphere with radius agree on every point, the two principal curvatures coincide:
  • Given the curved surface of a straight circular cylinder, with the base circle radius. In this case, the principal curvatures at each point of the surface have the values ​​0 ( tangential direction parallel to the axis of the cylinder ) and ( tangential direction perpendicular to the axis of the cylinder).
  • The same applies to cones and more generally, for developable surfaces ( Torsen ).
  • Consider an ellipsoid with semi-axes, and. In the end points ( vertices ) of the semi-axis, the principal curvatures are equal and.

Properties

  • The two main curvatures are the eigenvalues ​​of the vineyard illustration.
  • If the two principal curvatures match, then each is tangential main direction of curvature. Otherwise, there is exactly one principal curvature directions to each of the two principal curvatures. The two are orthogonal to each other.
  • Restricting the second fundamental form on the unit circle in the tangent plane, then the resulting function has the principal curvatures as extreme values ​​.
  • The Gaussian curve is the product of the main curvatures:
  • The mean curvature is the arithmetic average of the principal curvatures:
  • Is the Gaussian curvature and the mean curvature is known, the principal curvatures result as solutions of the quadratic equation
  • For each tangential direction can be the normal curvature expressed by the two principal curvatures:

Classification of surface points

A point of a surface is called [K 1]

  • Elliptic point if it is, so if both principal curvatures have the same sign;
  • Hyperbolic point if it is, so the signs are opposite;
  • Parabolic point if exactly one of the two principal curvatures is zero;
  • Flat point if and only if;
  • Umbilical point if and only if.

An elliptical umbilical point is also referred to as the actual umbilicus. A non- elliptical navel point is a flat point.

In elliptic points is the Gaussian curvature is positive ( ). This is the case when the centers of the circles of curvature of the normal sections are through both main directions on the same side of the surface, for example on the surface of an ellipsoid or ideological in double-curved shell structures such as domes. Hyperbolic points in the centers of the (main) bending two circuits, however, are on different sides of the surface as a saddle surface. The Gaussian curvature is negative there (). In parabolic points, such as on a cylindrical surface, or in flat points, the Gaussian curvature is zero.

The Dupin indicatrix in an elliptical point is an ellipse ( in an elliptical navel point, a circle ), in a hyperbolic point of a hyperbola and a parabolic point of a pair of parallel lines.

Related regular surfaces, made ​​entirely of umbilical points are subsets of a plane or a sphere. [K 2] [ dC 1]

Are on an open neighborhood of a point two vector fields given independently in linear, so there is a parametrization of a neighborhood of such that the vector fields are tangent to the coordinate lines. [ DC 2] If there is no umbilical point, so there is a parametrization an environment, so that the lines of curvature are coordinate lines, i.e., tangential to the main orthogonal directions. ( In an umbilical point is any direction main direction. ) In the vicinity of a hyperbolic point there is always a parametrization so that the coordinate lines are asymptotic, ie have vanishing normal curvature.

Plane (geometry)
378607
de