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8.1.3.5 Focal curves and surfaces

For a parametric space curve $ {\bf r}(t)$ , a focal curve or an evolute, shown in Fig. 8.5, is defined as
$\displaystyle {\bf f} (t) = {\bf r}(t) + {\bf n}(t)/\kappa(t)\;,$     (8.15)

where $ {\bf n}(t)$ is a unit principal normal vector and $ \kappa (t)$ is a nonzero curvature of the curve. Thus the focal curve or evolute of a curve is the locus of its centers of curvature. Focal surfaces [146,145] can be constructed in a similar way by using the principal curvature functions of the given surface. For a given parametric surface patch $ {\bf r}(u,v)$ the two associated focal surfaces are defined as
$\displaystyle {\bf f} (u,v) = {\bf r} (u,v) + {\bf N}(u,v)/\kappa(u,v)\;,$     (8.16)

where $ {\bf N}(u,v)$ is a unit surface normal and $ \kappa(u,v)$ is a nonzero principal curvature ( $ \kappa_{min}(u,v)$ or $ \kappa_{max}(u,v)$ ). Here sign convention (a) (see Fig. 3.7 (a)) is employed for both (8.15) and (8.16). When sign convention (b) is assumed, we simply replace the plus sign by the minus sign.

Figure 8.5: Focal curve. The thin curve is the focal curve of the thick curve
\begin{figure}\centerline{ \psfig{file=fig/vase_focal.ps,height=3.5in} }\end{figure}
Focal curves and surfaces provide another method for visualizing curvature. An important application of focal curves is in testing the curvature continuity of surfaces across a common boundary. Curves on the focal surfaces of each surface, which are at the common boundary, $ {\bf f} (t) = {\bf r} (u(t),v(t)) + {\bf N}(u(t),v(t))/\kappa(u(t),v(t))$ , where $ {\bf r}= {\bf r} (u(t),v(t))$ is the common boundary curve (or linkage curve) on the progenitor surface $ {\bf r}={\bf r}(u,v)$ , can be compared to determine the curvature continuity of the surfaces. According to the linkage curve theorem by Pegna and Wolter [305] two surfaces joined with first-order or tangent plane continuity along a first-order continuous linkage curve can be shown to be second-order smooth on the linkage curve if the normal curvatures along the linkage curve on each surface agree in one direction other than the tangent direction to the linkage curve. Comparison of focal surfaces allows for a visual assessment of the accuracy for the second-order contact of two patches along their linkage curve. Namely if the two focal curve point sets (defined via $ \kappa_{min}$ and $ \kappa_{max}$ ) of one surface patch agree along the linkage curve with the two corresponding focal curve point sets of the adjacent patch then these surface patches have curvature continuous surface contact along the linkage curve. The deviation of the corresponding focal curves indicates the amount of discontinuity of curvature of both adjacent patches along the linkage curve.

next up previous contents index
Next: 8.1.3.6 Orthotomics Up: 8.1.3 Second-order interrogation methods Previous: 8.1.3.4 Surface curvatures and   Contents   Index
December 2009