If we define a curve in parameter space, (and thus on the surface) and restrict the normal field to the curve, then we get a function of normals along the curve . The derivative is precisely , where . Futhermore, given a parameterization of the surface , we have a basis for vectors in the tangent plane and , can be represented in this basis:
The shape operator can be expressed in terms of the parameterization by invoking it on the tangent to a curve :
Substituting (2.41)-(2.42) into (2.50)-(2.51) generates a set of relations between , , and , for example, . When solved, these give
Copyright © 2005 Adrian Secord. All rights reserved.