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) |
