The negative derivative
(1)
of the unit normal vector field of a surface is called the shape operator (or Weingarten map or second fundamental tensor). The shape operator is an extrinsic curvature, and the Gaussian curvature is given by the determinant of . If is a regular patch, then
At each point on a regular surface , the shape operator is a linear map
(4)
The shape operator for a surface is given by the Weingarten equations.
See alsoCurvature,
Fundamental Forms,
Weingarten Equations Explore with Wolfram|Alpha ReferencesGray, A. "The Shape Operator," "Calculation of the Shape Operator," and "The Eigenvalues of the Shape Operator." ยง16.1, 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. Referenced on Wolfram|AlphaShape Operator Cite this as:Weisstein, Eric W. "Shape Operator." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ShapeOperator.html
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