Curvature Parametric surface

the first , second fundamental forms of surface determine important differential-geometric invariants: gaussian curvature, mean curvature, , principal curvatures.

the principal curvatures invariants of pair consisting of second , first fundamental forms. roots κ1, κ2 of quadratic equation

det
(

i
i

−
κ

i

)
=
0
,

det

|




l
−
κ
e


m
−
κ
f




m
−
κ
f


n
−
κ
g




|

=
0.


{\displaystyle \det(\mathrm {ii} -\kappa \mathrm {i} )=0,\quad \det \left|{\begin{matrix}l-\kappa e&m-\kappa f\\m-\kappa f&n-\kappa g\end{matrix}}\right|=0.}

the gaussian curvature k = κ1κ2 , mean curvature h = (κ1 + κ2)/2 can computed follows:

k
=



l
n
−

m

2




e
g
−

f

2





,

h
=



e
n
−
2
f
m
+
g
l


2
(
e
g
−

f

2


)



.


{\displaystyle k={ln-m^{2} \over eg-f^{2}},\quad h={en-2fm+gl \over 2(eg-f^{2})}.}

up sign, these quantities independent of parametrization used, , hence form important tools analysing geometry of surface. more precisely, principal curvatures , mean curvature change sign if orientation of surface reversed, , gaussian curvature entirely independent of parametrization.

the sign of gaussian curvature @ point determines shape of surface near point: k > 0 surface locally convex , point called elliptic, while k < 0 surface saddle shaped , point called hyperbolic. points @ gaussian curvature 0 called parabolic. in general, parabolic points form curve on surface called parabolic line. first fundamental form positive definite, hence determinant eg − f positive everywhere. therefore, sign of k coincides sign of ln − m, determinant of second fundamental.

the coefficients of first fundamental form presented above may organized in symmetric matrix:

f

1


=


[



e


f




f


g



]


.


{\displaystyle f_{1}={\begin{bmatrix}e&f\\f&g\end{bmatrix}}.}

and same coefficients of second fundamental form, presented above:

f

2


=


[



l


m




m


n



]


.


{\displaystyle f_{2}={\begin{bmatrix}l&m\\m&n\end{bmatrix}}.}

defining matrix



a
=

f

1


−
1



f

2




{\displaystyle a=f_{1}^{-1}f_{2}}

, principal curvatures κ1 , κ2 eigenvalues of a.

now, if v1=(v11,v12) eigenvector of corresponding principal curvature κ1, unit vector in direction of







t
→




1


=

v

11






r
→




u


+

v

12






r
→




v




{\displaystyle {\vec {t}}_{1}=v_{11}{\vec {r}}_{u}+v_{12}{\vec {r}}_{v}}

called principal vector corresponding principal curvature κ1.

accordingly, if v2=(v21,v22) eigenvector of corresponding principal curvature κ2, unit vector in direction of







t
→




2


=

v

21






r
→




u


+

v

22






r
→




v




{\displaystyle {\vec {t}}_{2}=v_{21}{\vec {r}}_{u}+v_{22}{\vec {r}}_{v}}

called principal

vector corresponding principal curvature κ2.