9.2 Lines of curvature near umbilics

It is easily verified from (9.1) that , and simultaneously vanish at the umbilics. Therefore for all , , (3.41) is satisfied, and hence we cannot determine the direction of the lines of curvature which pass through the umbilic. In this section we investigate the pattern of the lines of curvature near generic umbilics. Generic umbilics are stable with respect to small perturbations of the function representing the surface, while non-generic umbilics are unstable [24,364,397]. Darboux [71] has described three generic features of lines of curvature in the vicinity of an umbilic. The three generic features are called star, (le) monstar and lemon based on the pattern of the net of lines of curvature. Color Plate A.7 illustrates these three patterns of the net of lines of curvature at the umbilic. The red solid line corresponds to the maximum principal curvature lines and the dotted blue line corresponds to minimum principal curvature lines, where convention (b) of sign of normal curvature is used (see Fig. 3.7 (b) and Table 3.2). Three lines of curvature pass through the umbilic for monstar and star, while only one passes for the lemon. The criterion distinguishing monstar from star is that all three directions of lines of curvature through an umbilic are contained in a right angle, whereas in the star case they are not contained in a right angle. There are no other patterns except for non-generic cases. An example of a non-generic umbilic can be offered by the two poles of a convex closed surface of revolution [165]. Figure 9.1 (a) shows the non-generic umbilic of a paraboloid of revolution which has an umbilic that infinite number of lines of curvature pass through. If we perturb a coefficient in the function representing the surface slightly to corresponding to an elliptic paraboloid then the non-generic umbilic splits into two lemon-type generic umbilics as shown in Fig. 9.1 (b).

Consider a surface in Monge form

where is a smooth function, i.e. it has continuous derivatives up to order three. We can Taylor expand the z component of the surface as in (8.73). Suppose the surface has an umbilic at the origin and its tangent plane coincides with the plane, then it is apparent that . Evaluating the coefficients of the first and second fundamental forms of the explicit surface at the origin, which are given in (3.63) through (3.65), we obtain

Then it is apparent from (9.1) that and . Consequently we can rewrite (8.73) into a simpler form:

From (9.5), we can observe that the equation of the surface near the umbilic is governed by the cubic form

where

Note that , , and vanish for a paraboloid of revolution at .

Figure 9.1: Lines of curvature of paraboloid of revolution and elliptic paraboloid. Solid lines and dotted lines represent maximum and minimum principal curvature lines respectively: (a) paraboloid of revolution ( ) has a non-generic umbilic at (0,0,0), (b) elliptic paraboloid ( ) has two lemon-type umbilics at (0, 0.1890,0.0357) (adapted from [257])

To study the behavior of the umbilics we can express (9.6) in polar coordinates and for a fixed radius [24]:

It can be easily verified that . Therefore the cubic function is an antisymmetric function of . The roots of will give the angles where local maxima and minima of may occur around the umbilic, depending on the multiplicity of the roots. When there are three distinct roots, each of the roots gives the local extremum. When there are two equal roots, the double roots will provide neither maxima nor minima, however the single root gives an extremum. When there are three equal roots, the triple roots give an extremum. Since it is an antisymmetric function, maxima and minima of occur on the same straight line which passes through the umbilic. Differentiating (9.8) with respect to and setting the equation equal to zero yields

When one of the roots of (9.9) is or then must be zero, and when or then must be zero. Conversely we can say that when one of the roots is or , and when one of the roots is or . Consequently when , we can divide (9.9) by resulting in

where . Similarly when , we can divide (9.9) by resulting in

where . These cubic equations may be reduced by the substitution

to the normal form [403]

where

The solutions to the cubic equation are given by:

• When ; there are three distinct roots, one is real root and the other two are conjugate complex roots. The real root gives a function extremum and is given by
  
  
  
• When ; there are three real roots at least two of which are equal and are given by
  
  
  where the upper sign is to be used if is positive and the lower sign if is negative. Therefore there is at most one root which will provide a function extremum. This is a non-generic case, since small perturbation will yield the case either above or below.
• When ; there are three unequal real roots, which provide three function extrema, and are given by
  
  
  where and the upper sign is to be used if is positive and the lower if is negative.

Consequently there is either one single angle (lemon) or three different angles (star, monstar) corresponding to one maximum opposite one minimum or three maxima opposite three minima for generic case. Corresponding to these angles there are lines of curvature either one or three passing through the umbilics.

Another way of classifying an umbilic is to compute the index around it [24,364]. The lemon and monstar have the same index , while the star has the index . The index is defined as an amount of rotation that a straight line tangent to lines of curvature experiences when rotating in the counterclockwise direction around a small closed path around the umbilic. To compute the index of the umbilic, we can evaluate the angle , which is the angle of principal direction, at points along a boundary curve which surrounds the umbilic. The angle is obtained by using the first of (3.41) (see Table 3.2 for sign convention) as

where . Since can also be obtained from the second equation of (3.41) we also get

If both and are zero or small in absolute value, we use (9.22) otherwise we use (9.21), or if we can invert (9.21) and solve for . Consequently the index can be computed by

where

and is the modulo operator. It is used to account for the first point which is also the last point at which the direction field is evaluated. For the examples in this book, 20 points per boundary curve were adequate for estimation of the index. Fig. 9.2 illustrates the direction field of the maximum principal curvature around the star, monstar and lemon type umbilics.

Figure 9.2: Direction field near umbilics: (a) star-type, (b) monstar-type, (c) lemon-type (adapted from [257])