be a regular surface with points in the tangent
Then the first fundamental form is the inner product
of tangent vectors,
The first fundamental form satisfies
The first fundamental form (or line element) is given
explicitly by the Riemannian metric
It determines the arc length of a curve on a surface.
The coefficients are given by
The coefficients are also denoted , , and . In curvilinear
coordinates (where ), the quantities
are called scale factors.
See alsoFundamental Forms
, Third Fundamental Form
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ReferencesGray, 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|AlphaFirst Fundamental Form
Cite this as:
Weisstein, Eric W. "First Fundamental Form."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FirstFundamentalForm.html