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
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
Copyright © 2005 Adrian Secord. All rights reserved.