Gaussian curvature

In the theory of surfaces in three-dimensional Euclidean space (), an area of differential geometry, the Gaussian curvature ( the Gaussian curvature ), named after the mathematician Carl Friedrich Gauss, the principal curvature term in addition to the mean curvature.

Definition

Given a regular surface and a point in this area. The Gaussian curvature of the surface at this point is the product of the two principal curvatures and.

In this case, and the two main radii of curvature.

Examples

  • In the case of a sphere (surface ) with a radius of the Gaussian curvature is given by.
  • At an arbitrary point on the curved surface of a straight circular cylinder, the Gaussian curvature is equal to 0
  • At an arbitrary point on the curved surface of a straight circular cone, the Gaussian curvature is equal to 0
  • Is a graph of on - plane, the Gaussian curvature calculated by the formula

Where the subscripts denote partial derivatives.

Properties

  • In elliptic points is the Gaussian curvature is positive ( ), negative in hyperbolic points ( ) and parabolic points or flat points it disappears.
  • Are, respectively, the coefficients of the first and second fundamental form, the following formula applies:
  • The Gaussian curvature depends only on the intrinsic geometry of the given surface (see Theorema egregium of CF Gauss ). This theorem is a corollary of the:
  • Formula of Brioschi:
  • Another formula for the Gaussian curvature:
  • In the case of an orthogonal parametrization (), this formula reduces to
  • If the surface is parameterized isothermally, ie it applies and is then written as
  • If the surface is given as the zero set of a function with regular value, then the Gaussian curvature calculated from the formula
  • A relationship between the Gaussian curvature of a surface and the geodesic curvature of the corresponding boundary curve gives the Gauss -Bonnet.
Gradient Adjugate matrix Hessian matrix
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