There are three types of so-called fundamental forms. The most important are the first and second (since the third
can be expressed in terms of these). The fundamental forms are extremely important and useful in determining the
metric properties of a surface, such as Line Element, Area Element, Normal Curvature,
Gaussian Curvature, and Mean Curvature. Let be a Regular Surface with
points on the Tangent Space of . Then the first fundamental form is the
Inner Product of tangent vectors,

(1) |

(2) |

(3) |

The first and second fundamental forms satisfy

(4) | |||

(5) |

and so their ratio is simply the Normal Curvature

(6) |

(7) |

The first fundamental form (or Line Element) is given explicitly by the Riemannian Metric

(8) |

(9) | |||

(10) | |||

(11) |

The coefficients are also denoted , , and . In Curvilinear Coordinates (where ), the quantities

(12) | |||

(13) |

are called Scale Factors.

The second fundamental form is given explicitly by

(14) |

(15) | |||

(16) | |||

(17) |

and are the Direction Cosines of the surface normal. The second fundamental form can also be written

(18) | |||

(19) | |||

(20) |

where is the Normal Vector, or

(21) | |||

(22) | |||

(23) |

**References**

Gray, A. ``The Three Fundamental Forms.'' §14.6 in *Modern Differential Geometry of Curves and Surfaces.*
Boca Raton, FL: CRC Press, pp. 251-255, 259-260, 275-276, and 282-291, 1993.

© 1996-9

1999-05-26