Second fundamental form (shape operator)

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

$\displaystyle II_p(v) = -\ensuremath{\langle dN_p(v), v \rangle}.$ (2.30)

Recall that $ dN_p(v)$ is the change in surface normal in the direction of $ v$ at a point $ p$ . 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 $ \alpha(s)$ on the surface. The field of normals restricted to the curve is also a function of $ s$ : $ N(s) = N \circ \alpha(s)$ . Since the normal is always perpendicular to the tangent plane, we have $ \ensuremath{\langle N(s), \alpha'(s) \rangle} = 0$ , which we can differentiate to obtain $ \ensuremath{\langle N'(s), \alpha'(s) \rangle} + \ensuremath{\langle N(s), \alpha''(s) \rangle} = 0$ . Therefore

$\displaystyle II_p(\alpha'(0))$ $\displaystyle = -\ensuremath{\langle dN_p(\alpha'(0)), \alpha'(0) \rangle}$ (2.31)
  $\displaystyle = -\ensuremath{\langle N'(0), \alpha'(0) \rangle}$ (2.32)
  $\displaystyle = \ensuremath{\langle N(0), \alpha''(0) \rangle}$ (2.33)
  $\displaystyle = \ensuremath{\langle N, kn \rangle}(p)$ (2.34)
  $\displaystyle = k_n(p)$ (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.