which has four umbilics [320].
There exists an analogy between normal curvature and stress in elasticity theory [185]. For 2-D problems for example, it is well known that whatever the state of stress at a point, there will always be two orthogonal directions through the point in each of which the shear stress is zero. These two directions are called the axes of principal stress. The curve which lies along one of the axes of principal stress at all its points is called line of principal stress. Such lines form an orthogonal net. The point where two principal stresses are equal is called isotropic point. The state of stress at such point is that of a radial compression or tension, uniform in all directions. The lines of principal stress and isotropic points are analogous to the lines of curvature and umbilics, respectively.
There are a number of papers which deal with lines of curvature. Martin [265] introduced so called the principal patches whose sides are lines of curvature for use in geometric modeling. Principal patches can be created by imposing two conditions to the boundary curves known as position and frame matching [265,290]. Among the principal patches, Dupin's cyclide patches whose lines of curvature are all circular arcs are used for blending surface applications (see Pratt [332], Dutta et al. [83]).
Beck et al. [22], Farouki [96,98], Hosaka [173], Maekawa et al. [257] provide a method to construct a net of lines of curvature on a B-spline surface. Lines of curvature are of considerable importance to plate-metal-based manufacturing [279]. When a sheet is to be shaped by rolling, then it is fed into the rolls according to a principal direction and the rolls are adjusted according to the principal curvature.
The generic features of the lines of curvature near an umbilic are
fully discussed in classic work by Darboux (
) [71],
and more recently by Porteous [320,321,322],
Maekawa et al. [257], and Gutierrez and Sotomayor
[143]. Berry and
Hannay [24] calculate the average density of umbilics for a
surface whose deviation from a plane is specified by a Gaussian random
surface, and showed the rarity of the monstar pattern. Maekawa
and Patrikalakis
[255] and Maekawa [246] describe a robust
computational method to locate all isolated umbilics on a polynomial
parametric surface and investigate the generic features of
umbilics and the behavior of lines of curvature which pass through an
umbilic. In computer vision, Brady et al.
[37] compute the lines of curvature and regions of umbilics
from range images. Sander and Zucker
[364] extracted umbilics from an image by computing the
index of the principal direction fields. Sinha and Besl [397]
compute the lines of curvature from a range image and construct a
quadrilateral mesh except at the umbilics.
A non-flat umbilic occurs at an elliptic point, while it never occurs at
a hyperbolic point. From (3.31),
where we assume convention (b) (see Fig. 3.7
(b) and Table
3.2), it is apparent that at an umbilic 
and 
are proportional because
, and hence we have the
following relation at the umbilic
. At umbilical points only, the principal directions are
indeterminate and the net of lines of curvature may have singular
properties. The lines of curvature depend only on the shape of the
surface, and not the parametrization. Lines of curvature provide a
method to describe the variation of principal curvatures across a
surface. Lines of curvature can be obtained by integrating
(3.41), which will be discussed further in Sect.
9.4.
In this chapter we employ sign convention (b) (see Fig. 3.7 (b) and Table 3.2) for the normal curvature.
