Page 375 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes We showed that the normal curvature k can be expressed as
N
k () = H + Acos 2 + B sin 2
N
 
over the orthonormal frame (e , e ).
1
2
A B
We also showed that the angle  such that cos 2 = C and sin 2 = C , plays a special role.
0
0
0
Indeed, it determines one of the principal directions.
   
If we rotate the basis (e , e ) and pick a frame (f , f ) corresponding to the principal directions,
2
2
1
1
we obtain a particularly nice formula for k . Indeed, since A = C cos 2 and B = C sin 2 , letting
N
0
0
=  –  , we get
0
k () = k cos  + k sin .
2
2
N
1
2

Thus, for any unit vector t expressed as
  
t = cos  1 f + sin  2 f
with respect to an orthonormal frame corresponding to the principal directions, the normal
curvature k () is given by
N
Euler’s formula (1760):
kN() = k cos  + k sin .
2
2
1
2
Recalling that EG – F is always strictly positive, we can classify the points on the surface
2
depending on the value of the Gaussian curvature K, and on the values of the principal curvatures
k and k (or H).
1
2
Definition 2: Given a surface X, a point p on X belongs to one of the following categories:
(1) Elliptic if LN – M > 0, or equivalently K > 0.
2
(2) Hyperbolic if LN – M < 0, or equivalently K < 0.
2
(3) Parabolic if LN – M = 0 and L + M + N > 0, or equivalently K = k k = 0 but either k  0
2
2
2
2
1
1 2
or k  0.
2
(4) Planar if L = M = N = 0, or equivalently k = k = 0.
1
2
Furthermore, a point p is an umbilical point (or umbilic) if K > 0 and k = k .
2
1
At an elliptic point, both principal curvatures are non-null and have the same sign. For

example, most points on an ellipsoid are elliptic.
At a hyperbolic point, the principal curvatures have opposite signs. For example, all

points on the catenoid are hyperbolic.
At a parabolic point, one of the two principal curvatures is zero, but not both. This is

equivalent to K = 0 and H  0. Points on a cylinder are parabolic.
At a planar point, k = k = 0. This is equivalent to K = H = 0. Points on a plane are all planar
 1 2
points! On a monkey saddle, there is a planar point. The principal directions at that point
are undefined.
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