Second Fundamental Form

The second fundamental form of a regular surface is the symmetric bilinear form on the tangent space given, for tangent vectors , by

(1)

where is the shape operator. The second fundamental form satisfies

(2)

for any nonzero tangent vector.

The second fundamental form is given explicitly by

(3)

where

			(4)
			(5)
			(6)

and are the direction cosines of the surface normal. The second fundamental form can also be written

			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)
			(14)

where is the normal vector, is a regular patch, and and are the partial derivatives of with respect to parameters and , respectively, or

			(15)
			(16)
			(17)