Saddle surface

A saddle surface is an area referred to in the geometry, which is opposite in the two main directions - i.e. anticlastic - is curved. Gaussian curvature is your negative.

Their name comes from the horse saddle and the saddle on the ground, which also represents a transition between two mountains and two valleys.

Examples of saddle surfaces

A well-known saddle surface is a hyperbolic paraboloid. Such a surface arises from the fact that it connects equally opposite edges of a spatial quadrangle by threads. Such a surface can therefore be generated by moving a straight line in space (she is a ruled surface ). Another generation of a hyperbolic paraboloid is to have as the locus of all points equidistant from two mutually skew lines in space.

Other special cases are minimal surfaces, in which opposite the two principal curvatures are equal.

Geometric properties

The angle sum of a triangle on a saddle surface is - as opposed to a spherical triangle or generally a triangle on a positively curved surface - less than 180 ° ( see accompanying diagrams).

The Gaussian curvature of the saddle surface is negative, that positive effect on sphere or ellipsoid. Therefore, the geometry differs on that surface of Euclidean geometry in the plane containing the curvature is zero.