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First Fundamental Form


Let M be a regular surface with v_(p),w_(p) points in the tangent space M_(p) of M. Then the first fundamental form is the inner product of tangent vectors,

 I(v_(p),w_(p))=v_(p)·w_(p).
(1)

The first fundamental form satisfies

 I(ax_u+bx_v,ax_u+bx_v)=Ea^2+2Fab+Gb^2.
(2)

The first fundamental form (or line element) is given explicitly by the Riemannian metric

 ds^2=Edu^2+2Fdudv+Gdv^2.
(3)

It determines the arc length of a curve on a surface. The coefficients are given by

E=||x_u||^2=|(partialx)/(partialu)|^2
(4)
F=x_u·x_v=(partialx)/(partialu)·(partialx)/(partialv)
(5)
G=||x_v||^2=|(partialx)/(partialv)|^2.
(6)

The coefficients are also denoted g_(uu)=E, g_(uv)=F, and g_(vv)=G. In curvilinear coordinates (where F=0), the quantities

h_u=sqrt(g_(uu))=sqrt(E)
(7)
h_v=sqrt(g_(vv))=sqrt(G)
(8)

are called scale factors.


See also

Fundamental Forms, Second Fundamental Form, 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

First Fundamental Form

Cite this as:

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

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