Second fundamental form (shape operator)

The second fundamental form is similar to the first fundamental form (sec:FirstFundamentalForm), except that it relates a vector in the tangent plane to the change in normal in the direction of . Specifically, the second fundamental form or shape operator at a point is defined as

(2.30) |

Recall that is the change in surface normal in the direction of at a point . Similarly to the case of curves (sec:CurvatureCurves), the change in surface normals encodes information about the curvature of the surface.

To see how this might be true, consider a regular arc length-parameterized curve on the surface. The field of normals restricted to the curve is also a function of : . Since the normal is always perpendicular to the tangent plane, we have , which we can differentiate to obtain . Therefore

(2.31) | ||

(2.32) | ||

(2.33) | ||

(2.34) | ||

(2.35) |

So along a curve on the surface, the second fundamental form at a point gives the normal curvature to an arc length-parameterized curve.

Copyright © 2005 Adrian Secord. All rights reserved.