§ Second fundamental form
- Let be a (local) parametrization of the surface. Taylor expand . we get:
- We must get such a taylor expansion since our output is 1D (a real number), inputs are which are 3D vectors, and the infinitesimals must be linear/tensorial. These are the only possible contractions we can make.
- So, the second degree part can be written as:
- the matrix in the middle, or the quadratic form is the second fundamental form.
§ Classical geometry
- Let be a (local) parametrization of the surface.
- At each point on the surface within the local parametrization, we get tangent vectors which span the tangent space at
- These define a unique normal vector at each point on the surface. This gives us a normal field.
- The coefficient of the second fundamental form project the second derivative of the function onto the normals. So they tell us how much the function is escaping the surface (ie, is moving along the normal to the surface) in second order.
- Recall that this is pointless to do for first order, since on a circle, tangent is perpendicular to normal, so any dot product of first order information with normal will be zero.
- Alternatively, first order information lies on tangent plane, and the normal is explicitly constructed as perpendicular to tangent plane, so any dot product of first order info with normal is zero.
- We can only really get meaningful info by dotting with normal at second order.
- So we get that , , and , where we define via second fundamental form
§ Proof of equivalence between 2nd fundamental form and geometry
§ Principal curvature
- take a point . Consider the normal to the surface at the point, .
- Take any normal plane: a plane which contains . This plane (which is normal to the surface, since it contains the normal) intersecs the surface at a curve (intuitively, since a plane in 3D is defined by 1 eqn, intersection with plane imposes 1 equation on the surface, cutting it down to 1D).
- The curvature of this curve (normal plane intersection surface ) at point is the normal curvature of the normal plane .
- The maximum and minimum such normal curvatures at a point (max, min taken across all possible normal planes ) are the principal curvatures.
§ Shape operator has principal curvatures as eigenvalues
§ Shape operator in index notation
- Let be tangent vectors at point , be normal to surface at point . The shape operator is determined by the equation: