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Shape Operator

The negative derivative

S(v)=-D_(v)N
(1)

of the unit normal N vector field of a surface is called the shape operator (or Weingarten map or second fundamental tensor). The shape operator S is an extrinsic curvature, and the Gaussian curvature is given by the determinant of S. If x:U->R^3 is a regular patch, then

S(x_u)=-N_u
(2)
S(x_v)=-N_v.
(3)

At each point p on a regular surface M subset R^3, the shape operator is a linear map

S:M_(p)->M_(p).
(4)

The shape operator for a surface is given by the Weingarten equations.

See also

Curvature, Fundamental Forms, Weingarten Equations

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References

Gray, A. "The Shape Operator," "Calculation of the Shape Operator," and "The Eigenvalues of the Shape Operator." §16.1, 16.3, and 16.4 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 360-363 and 367-372, 1997.Reckziegel, H. In Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 30, 1986.

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Shape Operator

Cite this as:

Weisstein, Eric W. "Shape Operator." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ShapeOperator.html

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