9.4 Integration of lines of curvature

A line of curvature is a curve on a surface that has tangents which are principal directions at all of its points as we discussed in Sect. 3.4. The principal directions at a given point are those directions for which the normal curvature takes on minimum and maximum values. If the point is not an umbilic the principal directions are orthogonal. A line of curvature indicates a directional flow for the maximum or the minimum curvature across the surface. It is advantageous to express the curvature line with an arc length parametrization as . Every principal curvature direction vector must fulfill (3.41). Hence from the first equation of (3.41) (see Table 3.2) we get

where is an arbitrary nonzero factor. At first sight, one may expect to obtain the lines of curvature by integrating (9.47). However the subsequent considerations show that a simple integration of (9.47) is generally not sufficient to compute the principal curvature lines even in situations where one does not encounter an umbilic. Namely there may occur several problems, including cases and below, and the criterion in (9.52) may be used to control the orientation while integrating along the curvature line. Since a principal curvature direction vector must also fulfill the second equation of (3.41) we also get

The solutions , of the first and the second equations of (3.41) are linearly dependent, because the system of linear equations given by (3.41) has a rank smaller than 2. It is possible:

Case A:

That the coefficients in one of the equations can both be zero while they are not both zero in the other equation.

Case B:

That both coefficients in one equation are small in absolute value while the other equation contains one coefficient which is large in absolute value.

Case is encountered more often than case . In case , using the equation with zero coefficients yields an incorrect result, because this equation does not contain enough information to find the principal curvature direction. In case , using the equation with the small coefficients may yield numerical inaccuracies which could be avoided by using the other equation. Alourdas [5] has developed an algorithm which makes the choice of the equation dependent on the size of the coefficients. Since is a common coefficient, if we solve (9.47) otherwise we solve (9.48).

We want to point out that also case may easily occur. Therefore one needs provisions in the algorithm which takes this into account. We give now a simple example illustrating case using a parabolic cylinder . Clearly the maximum principal curvature on the parabolic cylinder is zero everywhere. Also it is apparent that and , hence . Therefore and become zero, while , which can be seen by an easy computation or using the fact that a parabolic cylinder has no umbilics. Farouki [98] proved that one of the solutions ( , ) defining a principal direction (i.e. (9.47) or (9.48)) becomes indeterminate at a nonumbilic point if and only if the principal direction is tangent to a surface parameter line at that point as in this example.

It remains to determine factors and . If the curvature line is arc length parametrized, the first fundamental form provides the normalization condition

Substituting (9.47) into (9.49), is determined to be

Likewise is determined to be

The sign of or determines the direction in which the solution proceeds. Choosing a fixed sign for or does not guarantee that the vector would not change direction. The need to adjust the sign of or becomes even more obvious if one determines the principal curvature vector always by the numerically preferable equation in the system (3.41). The vectors obtained from (9.47) and (9.48) are linearly dependent but they do not need to have the same orientation.

The criterion which is employed in order to determine the sign of or is given by the following inequality

where is a curvature line represented by the parametric form = , and the superscript means evaluation at the previous time step during the integration of the curvature line. It is obvious that inequality (9.52) is true if and only if the tangent vector reverses direction because (9.52) says that the negative tangent vector of the preceding time step is closer to the new tangent vector than the positive tangent vector of the preceding time step. When inequality (9.52) is true, the sign of or should be changed to assure that the solution path does not reverse direction. Farouki [98] derives another criterion for preventing the reversal of integration direction.

We can trace the lines of curvature by integrating the initial value problem for a system of coupled nonlinear ordinary differential equations using standard numerical techniques [69,126] such as Runge-Kutta method or a more sophisticated variable stepsize and variable order Adams method. Starting points for lines of curvature passing through the umbilics are obtained by slightly shifting outwards in the directions given by (9.46) from the umbilic. Accuracy of the lines of curvature depends on the number of integrated points used to represent the contour line by straight line segments.