Principal Curvature - Line of Curvature

Line of Curvature

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.

In the vicinity of an umbilic the lines of curvature typically form one of three configurations star, lemon and monstar (derived from lemon-star). These points are also called Darbouxian Umbilics, in honor to Gaston Darboux, the first to make a systematic study in Vol. 4, p 455, of his Leçons (1896).

• configurations of lines of curvature near umbilics

• Lemon

• Monstar

• Star

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.