Page 376 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 30: Geodesic Curvature and Christoffel Symbols

Notes
Figure 30.1: A Monkey Saddle

For an umbilical point, we have k = k  0.
2
1
2
2
This can only happen when H – C = H + C, which implies that C = 0, and since C = A + B , we
have A = B = 0.
Thus, for an umbilical point, K = H . In this case, the function k is constant, and the principal
2
N
directions are undefined. All points on a sphere are umbilics. A general ellipsoid (a, b, c pairwise
distinct) has four umbilics. It can be shown that a connected surface consisting only of umbilical
points is contained in a sphere. It can also be shown that a connected surface consisting only of
planar points is contained in a plane.

A surface can contain at the same time elliptic points, parabolic points, and hyperbolic points.
This is the case of a torus.
The parabolic points are on two circles also contained in two tangent planes to the torus

(the two horizontal planes touching the top and the bottom of the torus on the following
picture).
The elliptic points are on the outside part of the torus (with normal facing outward),

delimited by the two parabolic circles.
The hyperbolic points are on the inside part of the torus (with normal facing inward).


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