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Concept
Second fundamental form
Summary
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 (read "two"). Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth immersed submanifold in a Riemannian manifold.
The second fundamental form of a parametric surface S in R3 was introduced and studied by Gauss. First suppose that the surface is the graph of a twice continuously differentiable function, z = f(x,y), and that the plane z = 0 is tangent to the surface at the origin. Then f and its partial derivatives with respect to x and y vanish at (0,0). Therefore, the Taylor expansion of f at (0,0) starts with quadratic terms:
and the second fundamental form at the origin in the coordinates (x,y) is the quadratic form
For a smooth point P on S, one can choose the coordinate system so that the plane z = 0 is tangent to S at P, and define the second fundamental form in the same way.
The second fundamental form of a general parametric surface is defined as follows. Let r = r(u,v) be a regular parametrization of a surface in R3, where r is a smooth vector-valued function of two variables. It is common to denote the partial derivatives of r with respect to u and v by ru and rv. Regularity of the parametrization means that ru and rv are linearly independent for any (u,v) in the domain of r, and hence span the tangent plane to S at each point. Equivalently, the cross product ru × rv is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors n:
The second fundamental form is usually written as
its matrix in the basis {ru, rv} of the tangent plane is
The coefficients L, M, N at a given point in the parametric uv-plane are given by the projections of the second partial derivatives of r at that point onto the normal line to S and can be computed with the aid of the dot product as follows:
For a signed distance field of Hessian H, the second fundamental form coefficients can be computed as follows:
The second fundamental form of a general parametric surface S is defined as follows.
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