Theorem 9.7.2. There is only one line of curvature that passes through each flat point on a line of flat points, and that line of curvature is orthogonal to the direction of the generator [251].
Proof: By Lemma 9.7.1 the developable surface is
expressed locally as a cubic cylinder at an ordinary inflection line
and more generally in the form of (9.95) at a higher
order contact line. If we rewrite (9.95) in terms of
polar coordinates by substituting
for a fixed radius
we obtain
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(9.97) |
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(9.98) |
If we denote
as the angle between the
axis and the
direction of the nonzero principal curvature in
parametric space,
can be evaluated as follows. Since the direction of the nonzero
principal curvature is orthogonal to the
generator (parallel to the local
axis), its direction is given by
and hence
.
We can trace the lines of curvature which pass through the flat
points of an inflection line by integrating the initial value problem
following the procedure described in Sect. 9.4.
The starting points are obtained by slightly
shifting outwards in the directions 0
and
from the flat points
or, equivalently, along the positive and negative local
axis.
In generic cases, umbilics are isolated [257]; thus an inflection line, which consists of a line of flat points, is non-generic and therefore unstable. In the following we give a couple of numerical examples that demonstrate the instability of the line of flat points along the inflection line with respect to perturbations.
The example surface is a degree (3-1) integral Bézier patch which is constructed by the method developed in Chalfant [50]. The control points are given by
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We gradually perturb the control points of the surface and observe the
behavior of the lines of curvature which pass through the inflection
line as we did in Sect. 9.6. Since the example is a
degree (3-1) patch, it has 8 control points. Each control point
consists of three Cartesian coordinates
, so there are 24
components to be perturbed. We gradually increase the perturbation by
increasing
in (9.73) from
to
in steps of
.
Figure 9.13 illustrates the behavior of the lines of
curvature when the control points are perturbed (
). We
can see from the figure that the entire inflection line, which
consists of a line of flat points, disappears. Hence there is no
singularity in the net of lines of curvature when a perturbation is
induced. The nonzero principal curvatures on both sides of the former
inflection line 9.6 meet at right
angles near the former inflection line and make a very sharp change in
direction (almost a right angle).