At flat umbilic points both principal curvatures are zero.Parabolic points generally lie in a curve separating elliptical and hyperbolic regions. A generic surface will not contain flat umbilic points.
The monkey saddle is one surface with an isolated flat umbilic. The lines of curvature or curvature lines are curves which are always tangent to a principal direction (they are integral curves for the principal direction fields). There will be two lines of curvature through each non-umbilic point and the lines will cross at right angles. These points are also called Darbouxian Umbilics (D 1, D 2, D 3) in honor of In the vicinity of an umbilic the lines of curvature typically form one of three configurations star, lemon and monstar (derived from lemon-star). Configurations of lines of curvature near umbilics.Gaston Darboux, the first to make a systematic study in Vol. In these figures, the red curves are the lines of curvature for one family of principal directions, and the blue curves for the other. When a line of curvature has a local extremum of the same principal curvature then the curve has a ridge point. These ridge points form curves on the surface called ridges. The ridge curves pass through the umbilics. For the star pattern either 3 or 1 ridge line pass through the umbilic, for the monstar and lemon only one ridge passes through.
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