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9.7.2 Lines of curvature near inflection lines

At flat points the principal directions are indeterminate and the orthogonal net of lines of curvature may have singular properties. In the following we investigate the pattern of the lines of curvature near a line of non-generic flat points.

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 $ X=r\cos\theta$ for a fixed radius $ r = \sqrt{X^2 + Y^2}$ we obtain

$\displaystyle h(\theta)=c\cos^{k+1}\theta\;,$     (9.97)

where $ c$ is a constant evaluated at a point on the line of flat points given by
$\displaystyle c = \frac{r^{k+1}}{(k+1)!}\left(\frac{\partial^{k+1}{\bf r}}{\partial u^{k+1}}\cdot{\bf N}\right)\left(\sqrt{\frac{G}{EG-F^2}}\right)^{k+1}\;.$     (9.98)

If $ k$ is even, $ h(\theta+\pi) = -h(\theta)$ , and $ h(\theta)$ is an antisymmetric function of $ \theta$ , whereas if $ k$ is odd $ h(\theta)$ is a symmetric function of $ \theta$ . The roots of $ \frac{dh}{d\theta}=0$ will give the angles where local maxima and minima of $ h(\theta)$ may occur around the flat point. The equation can be restricted to the range $ 0 \leq \theta < 2\pi$ without loss of generality. The roots are easily computed as $ \theta = 0$ , $ \frac{\pi}{2}$ , $ \pi$ and $ \frac{3\pi}{2}$ . Only $ \theta = 0$ and $ \theta=\pi$ (which coincide with the local $ x$ axis) give extrema, since $ \frac{d^2h\left(\frac{\pi}{2}\right)}{d\theta^2} = \frac{d^2h\left(\frac{3\pi}{2}\right)}{d\theta^2} = 0$ . Thus $ \theta = \frac{\pi}{2}$ and $ \theta=\frac{3\pi}{2}$ (which coincide with the local $ y$ axis) provide neither a maximum nor a minimum. In other words, a line of flat points is not a line of curvature. Consequently, there is only one line of curvature passing through each flat point and it is orthogonal to the direction of the generator. For an even $ k$ the lines of maximum/minimum principal curvature switch to lines of minimum/maximum principal curvature at the inflection line since $ h(\theta)$ is antisymmetric, while for odd $ k$ they remain the same, since $ h(\theta)$ is symmetric.

If we denote $ \phi$ as the angle between the $ u$ axis and the direction of the nonzero principal curvature in $ uv$ parametric space, $ \phi$ can be evaluated as follows. Since the direction of the nonzero principal curvature is orthogonal to the generator (parallel to the local $ x$ axis), its direction is given by $ {\bf r}_v \times({\bf r}_u \times {\bf r}_v ) = ({\bf r}_v \cdot {\bf r}_v) {\bf r}_u - ({\bf r}_v \cdot {\bf r}_u) {\bf r}_v = G{\bf r}_u - F {\bf r}_v$ and hence $ \phi = -\tan^{-1} \frac{F}{G}$ .

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 $ \pi$ from the flat points or, equivalently, along the positive and negative local $ x$ 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

$ {\bf b}_{00}$ =$ (0,0,0)^T$ , $ {\bf b}_{01}$ = $ (0.5, 0, 2)^T$ ,
$ {\bf b}_{10}$ = $ (1.8, 3, 0)^T$ , $ {\bf b}_{11}$ = $ (1.895, 2.325,2)^T$ ,
$ {\bf b}_{20}$ = $ (3.3,-2,1.5)^T$ , $ {\bf b}_{21}$ = $ (3.0575,-1.55,3.1625)^T$ ,
$ {\bf b}_{30}$ =$ (4,0,0)^T$ , $ {\bf b}_{31}$ = $ ( 3.6,0,2)^T$ .
The surface has an ordinary inflection line at $ u=0.5754$ , which has been computed by solving the degree 5 univariate polynomial equation (9.88). This surface has a net of lines of curvature which is shown in Fig. 9.12(a). Solid lines represent the lines of maximum principal curvature, while dotted lines represent the lines of minimum principal curvature. The inflection line is depicted with a dash dotted line. Figure 9.12(b) shows a magnification near the inflection line. We can observe that there is only one line of curvature that passes through a flat point orthogonal to the inflection line.
Figure 9.12: (a) Lines of curvature of developable surface with inflection, (b) magnification near inflection line (adapted from [257])
\begin{figure}\centerline{ \psfig{figure=fig/inf00_loc.PS,height=2.0in,width=2.0... ...ht=2.0in,width=2.0in} } \mbox{\hspace*{3cm} (a) \hspace*{5cm} (b)}\end{figure}
\begin{figure}\centerline{ \psfig{figure=fig/inf08_loc.PS,height=2.0in,width=2.0... ...fication near $u$=0.57 (adapted from \protect\cite{Maekawa96a})} } \end{figure}

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 $ x,\;y,\;z$ , so there are 24 components to be perturbed. We gradually increase the perturbation by increasing $ \zeta $ in (9.73) from $ 0.02$ to $ 0.08$ in steps of $ 0.02$ .

Figure 9.13 illustrates the behavior of the lines of curvature when the control points are perturbed ( $ \zeta = 0.08$ ). 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).


Footnotes

... line9.6
Once the control points are perturbed both principal curvatures may not be nonzero, but here we are referring to the nonzero principal curvature before perturbation.
next up previous contents index
Next: 10. Geodesics Up: 9.7 Inflection lines of Previous: 9.7.1 Differential geometry of   Contents   Index
December 2009