| 释义 |
Fundamental FormsThere are three types of so-called fundamental forms. The most important are the first and second (since the thirdcan be expressed in terms of these). The fundamental forms are extremely important and useful in determining themetric 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 theInner Product of tangent vectors,
 | (1) |
 , the second fundamental form is the symmetric bilinear form on the Tangent Space ,
 | (2) |
 is the Shape Operator. The third fundamental form is given by
 | (3) |
The first and second fundamental forms satisfy
and so their ratio is simply the Normal Curvature
 | (6) |
 | (7) |
 is the Mean Curvature and  is the Gaussian Curvature.
The first fundamental form (or Line Element) is given explicitly by the Riemannian Metric
 | (8) |
The coefficients are also denoted  ,  , and  . In Curvilinear Coordinates (where  ), the quantities
are called Scale Factors.
The second fundamental form is given explicitly by
 | (14) |
and  are the Direction Cosines of the surface normal. The second fundamental formcan also be written
where  is the Normal Vector, or
See also Arc Length, Area Element, Gaussian Curvature,Geodesic, Kähler Manifold,Line of Curvature, Line Element, Mean Curvature,Normal Curvature, Riemannian Metric, Scale Factor, Weingarten Equations
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.
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