Second fundamental form Parametric surface

the second fundamental form

i
i

=
l


d


u

2


+
2
m


d

u


d

v
+
n


d


v

2




{\displaystyle \mathrm {ii} =l\,{\text{d}}u^{2}+2m\,{\text{d}}u\,{\text{d}}v+n\,{\text{d}}v^{2}}

is quadratic form on tangent plane surface that, first fundamental form, determines curvatures of curves on surface. in special case when (u, v) = (x, y) , tangent plane surface @ given point horizontal, second fundamental form quadratic part of taylor expansion of z function of x , y.

for general parametric surface, definition more complicated, second fundamental form depends on partial derivatives of order 1 , two. coefficients defined projections of second partial derivatives of






r
→





{\displaystyle {\vec {r}}}

onto unit normal vector






n
→





{\displaystyle {\vec {n}}}

defined parametrization:

l
=




r
→




u
u


⋅



n
→



,

m
=




r
→




u
v


⋅



n
→



,

n
=




r
→




v
v


⋅



n
→



.



{\displaystyle l={\vec {r}}_{uu}\cdot {\vec {n}},\quad m={\vec {r}}_{uv}\cdot {\vec {n}},\quad n={\vec {r}}_{vv}\cdot {\vec {n}}.\quad }

like first fundamental form, second fundamental form may viewed family of symmetric bilinear forms on tangent plane @ each point of surface depending smoothly on point.