The Second Fundamental Form

Capture how surfaces curve in space

The Second Fundamental Form

A flat sheet and a rolled-up tube measure distances the same way, yet one is clearly bent. The second fundamental form captures that bending, direction by direction.

Normal Curvature Explorer

Click a point on the surface, then turn the direction slider. Watch κn change: red when the surface bends toward the normal, blue when it bends away.

Click on the surface to select a point
Direction (rotate with slider)
κn > 0 (bends toward normal)
κn < 0 (bends away)
Go deeper:The Coefficients L, M, N

While the first fundamental form measures distances within the surface, the second fundamental form captures how the surface bends in the surrounding 3D space.

It encodes the normal curvature — how much the surface curves in each direction — via three coefficients L, M, N that describe the rate of change of the normal vector.

L = ⟨Suu, N⟩
Curvature in u-direction
M = ⟨Suv, N⟩
Mixed curvature term
N = ⟨Svv, N⟩
Curvature in v-direction

These measure the component of the second derivatives that points in the normal direction — i.e., how fast the surface is “pulling away” from the tangent plane.

Go deeper:Normal Curvature
κn = II(v,v) / I(v,v)

For a unit tangent vector v = (du, dv), the normal curvature is:

κn = (L du² + 2M du dv + N dv²) / (E du² + 2F du dv + G dv²)
Go deeper:The Normal Section

The normal section in direction v is the curve formed by slicing the surface with a plane containing both v and the normal N.

The normal curvature κn equals the ordinary curvature of this planar curve. It's positive if the curve bends toward N, negative if away.

Go deeper:The Shape Operator (Weingarten Map)

The shape operator S is a linear map on the tangent plane defined by:

S(v) = -dN(v)

It measures how the normal vector N changes as you move in direction v. The second fundamental form is related by:

II(v, w) = ⟨S(v), w⟩ = I(S(v), w)
Go deeper:Preview: Principal Curvatures

The principal curvatures k1 and k2 are the maximum and minimum values of normal curvature κn over all directions.

They are the eigenvalues of the shape operator, and their corresponding eigenvectors are the principal directions. These will be explored in detail on the next page.

Key Takeaways

  • L, M, N are the coefficients of the second fundamental form.
  • Normal curvature κn = II(v,v) / I(v,v) measures bending in direction v.
  • The shape operator S = -dN captures how the normal changes.
  • The second form is extrinsic: it depends on how the surface sits in 3D.

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