TOPICS

# Fundamental Forms

There are three types of so-called fundamental forms. The most important are the first and second (since the third can be expressed in terms of these). The fundamental forms are extremely important and useful in determining the metric properties of a surface, such as line element, area element, normal curvature, Gaussian curvature, and mean curvature. Let be a regular surface with points in the tangent space of . Then the first fundamental form is the inner product of tangent vectors,

 (1)

For , the second fundamental form is the symmetric bilinear form on the tangent space ,

 (2)

where is the shape operator. The third fundamental form is given by

 (3)

The first and second fundamental forms satisfy

 (4) (5)

where is a regular patch and and are the partial derivatives of with respect to parameters and , respectively. Their ratio is simply the normal curvature

 (6)

for any nonzero tangent vector. The third fundamental form is given in terms of the first and second forms by

 (7)

where is the mean curvature and is the Gaussian curvature.

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

 (8)

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

 (9) (10) (11)

The coefficients are also denoted , , and . In curvilinear coordinates (where ), the quantities

 (12) (13)

are called scale factors.

The second fundamental form is given explicitly by

 (14)

where

 (15) (16) (17)

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

 (18) (19) (20) (21) (22) (23) (24) (25)

where is the normal vector (Gray 1997, p. 368), or

 (26) (27) (28)

(Gray 1997, p. 379).

Arc Length, Area Element, First Fundamental Form, Gaussian Curvature, Geodesic, Kähler Manifold, Line of Curvature, Line Element, Mean Curvature, Normal Curvature, Riemannian Metric, Scale Factor, Second Fundamental Form, Surface Area, Third Fundamental Form, Weingarten Equations

## Explore with Wolfram|Alpha

More things to try:

## 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. 368-371 and 380-382, 1997.

## Referenced on Wolfram|Alpha

Fundamental Forms

## Cite this as:

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