An intrinsic property of a space independent of the coordinate system used to describe it. The Gaussian curvature of a
Regular Surface in at a point p is formally defined as
is a Regular Patch, then the Gaussian curvature is given by
The Gaussian curvature is also given by
The Gaussian curvature and Mean Curvature satisfy
If p is a point on a Regular Surface
vectors to at p, then the Gaussian curvature of at p is related to the Shape Operator by
For a Sphere, the Gaussian curvature is . For Euclidean Space, the Gaussian curvature is . For Gauss-Bolyai-Lobachevsky Space, the Gaussian curvature is . A Flat Surface is a Regular Surface and special class of Minimal Surface on which Gaussian curvature vanishes everywhere.
A point p on a Regular Surface is classified based on the sign of as given in the following table (Gray 1993, p. 280), where is the Shape Operator.
A surface on which the Gaussian curvature is everywhere Positive is called Synclastic, while a surface on which is everywhere Negative is called Anticlastic. Surfaces with constant Gaussian curvature include the Cone, Cylinder, Kuen Surface, Plane, Pseudosphere, and Sphere. Of these, the Cone and Cylinder are the only Flat Surfaces of Revolution.
See also Anticlastic, Brioschi Formula, Developable Surface, Elliptic Point, Flat Surface, Hyperbolic Point, Integral Curvature, Mean Curvature, Metric Tensor, Minimal Surface, Parabolic Point, Planar Point, Synclastic, Umbilic Point
Geometry Center. ``Gaussian Curvature.''
Gray, A. ``The Gaussian and Mean Curvatures'' and ``Surfaces of Constant Gaussian Curvature.'' §14.5 and Ch. 19 in
Modern Differential Geometry of Curves and Surfaces.
Boca Raton, FL: CRC Press, pp. 279-285 and 375-387, 1993.
Gray, A. ``The Gaussian and Mean Curvatures'' and ``Surfaces of Constant Gaussian Curvature.'' §14.5 and Ch. 19 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 279-285 and 375-387, 1993.
© 1996-9 Eric W. Weisstein