Let 
be a regular surface with 
points in the tangent
space 
of 
.
For 
,
the second fundamental form is the symmetric bilinear form on the tangent
space 
,
 |
(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)
|
