In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by
II
The second fundamental form of a parametric surface in was introduced and studied by Gauss. First suppose that the surface is the graph of a twice continuously differentiable function,, and that the plane is tangent to the surface at the origin. Then and its partial derivatives with respect to and vanish at (0,0). Therefore, the Taylor expansion of f at (0,0) starts with quadratic terms:
z=L | x2 |
2 |
+Mxy+N
y2 | |
2 |
+higherorderterms,
and the second fundamental form at the origin in the coordinates is the quadratic form
Ldx2+2Mdxdy+Ndy2.
For a smooth point on, one can choose the coordinate system so that the plane is tangent to at, and define the second fundamental form in the same way.
The second fundamental form of a general parametric surface is defined as follows. Let be a regular parametrization of a surface in, where is a smooth vector-valued function of two variables. It is common to denote the partial derivatives of with respect to and by and . Regularity of the parametrization means that and are linearly independent for any in the domain of, and hence span the tangent plane to at each point. Equivalently, the cross product is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors :
n=
ru x rv | |
|ru x rv| |
.
The second fundamental form is usually written as
II=Ldu2+2Mdudv+Ndv2,
its matrix in the basis