§ Second fundamental form
- Let z=f(x,y) be a (local) parametrization of the surface. Taylor expand f. we get:
- f(x+dx,y+dy)=f(x,y)+dxTa+dyTb+dxTLdx+2dxTMdy+dyTNdy.
- We must get such a taylor expansion since our output is 1D (a real number), inputs are dx,dy 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:
[xy][LMMN][xy]
- the matrix in the middle, or the quadratic form II≡dxTLdx+2dxTMdy+dyTNdy is the second fundamental form.
§ Classical geometry
- Let z=f(x,y) be a (local) parametrization of the surface.
- At each point p≡(u,v) on the surface within the local parametrization, we get tangent vectors ru(p)≡(∂xf(x,y)p,rv(p)≡(∂yf(x,y))p which span the tangent space at p
- These define a unique normal vector n(p)≡ru(p)×rv(p) 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 f 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 L(p)=(∂x∂xf(x,y))(p)⋅N(p), M(p)=(∂x∂yf(x,y))(p), and N(p)=(∂y∂yf(x,y))(p), where we define L,M,N via second fundamental form
§ Proof of equivalence between 2nd fundamental form and geometry