The second fundamental form of a regular surface is the symmetric bilinear form on the tangent space given, for tangent vectors , by

where is the shape operator. The second fundamental form satisfies

for any nonzero tangent vector.

The second fundamental form is given explicitly by

where

and are the direction cosines of the surface normal. The second fundamental form can also be written

where is the normal vector, is a regular patch, and and are the partial derivatives of with respect to parameters and , respectively, or

See also

First Fundamental Form, Fundamental Forms, Initial Data Set, Shape Operator, Third Fundamental Form

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References

Gray, A. "The Three Fundamental Forms." §16.6 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 380-382, 1997.

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Second Fundamental Form

Cite this as:

Weisstein, Eric W. "Second Fundamental Form." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SecondFundamentalForm.html

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