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Second Fundamental Form


Let M be a regular surface with v_(p),w_(p) points in the tangent space M_(p) of M. For M in R^3, the second fundamental form is the symmetric bilinear form on the tangent space M_(p),

 II(v_(p),w_(p))=S(v_(p))·w_(p),
(1)

where S is the shape operator. The second fundamental form satisfies

 II(ax_u+bx_v,ax_u+bx_v)=ea^2+2fab+gb^2
(2)

for any nonzero tangent vector.

The second fundamental form is given explicitly by

 edu^2+2fdudv+gdv^2
(3)

where

e=sum_(i)X_i(partial^2x_i)/(partialu^2)
(4)
f=sum_(i)X_i(partial^2x_i)/(partialupartialv)
(5)
g=sum_(i)X_i(partial^2x_i)/(partialv^2),
(6)

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

e=-N_u·x_u
(7)
=N·x_(uu)
(8)
f=-N_v·x_u
(9)
=N·x_(uv)
(10)
=N_·x_(vu)
(11)
=-N_u·x_v
(12)
g=-N_v·x_v
(13)
=N·x_(vv),
(14)

where N is the normal vector, x:U->R^3 is a regular patch, and x_u and x_v are the partial derivatives of x with respect to parameters u and v, respectively, or

e=(det(x_(uu)x_ux_v))/(sqrt(EG-F^2))
(15)
f=(det(x_(uv)x_ux_v))/(sqrt(EG-F^2))
(16)
g=(det(x_(vv)x_ux_v))/(sqrt(EG-F^2)).
(17)

See also

First Fundamental Form, Fundamental Forms, 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.

Referenced on Wolfram|Alpha

Second Fundamental Form

Cite this as:

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

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