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

(1) |

If
is a Regular Patch, then the Gaussian curvature is given by

(2) |

(3) |

(4) |

(5) |

(6) |

(7) | |||

(8) |

Writing this out,

(9) |

The Gaussian curvature is also given by

(10) |

(11) |

where is the Levi-Civita Symbol, is the unit Normal Vector and is the unit Tangent Vector. The Gaussian curvature is also given by

(12) |

(13) |

The Gaussian curvature and Mean Curvature satisfy

(14) |

(15) |

If **p** is a point on a Regular Surface
and
and
are tangent
vectors to at **p**, then the Gaussian curvature of at **p** is related to the Shape Operator by

(16) |

(17) |

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.

Sign | Point |

Elliptic Point | |

Hyperbolic Point | |

but | Parabolic Point |

and | Planar Point |

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.

**References**

Geometry Center. ``Gaussian Curvature.'' http://www.geom.umn.edu/zoo/diffgeom/surfspace/concepts/curvatures/gauss-curv.html.

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

1999-05-25