8.1.3.3 Radial curves

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8.1.3.3 Radial curves

For a parametric space curve ${\bf r}(t)$ , a radial curve is defined as

$\displaystyle {\bf f} (t) = {\bf p} + {\bf n}(t)/\kappa(t)\;,$

(8.12)

where ${\bf p}$ is a fixed reference point in space, ${\bf n}(t)$ is the unit principal normal vector and $\kappa (t)$ is a nonzero curvature of the curve [227]. Here the sign convention (a) (see Fig. 3.7 (a)) is adopted for the curvature. For sign convention (b) we simply replace the plus sign with a minus sign. Radial curves are a method for visualizing curvature in a manner decoupled from the shape of the curve as illustrated in Fig. 8.4. When an inflection point is involved, as in Fig. 8.4, spikes in opposite directions occur. By viewing the radial curve, a curvature measure is visualized. However, the user is left without a direct reference as to the relationship between the curve point and the curvature value. The radial curve also provides a method for viewing the range of variation of the curve's normal vector. This range of variation can be obtained from the angular sector described by all rays emanating from p and passing through all points on the radial curve. Radial curves are useful in accessibility and interference analyses.

**Figure 8.4:** A radial curve of with a fixed reference point **${\bf p}=(-2,2,0)$** . The *thin curve* is the radial curve of the *thick curve* ( with a curvature plot)
$\begin{figure}\centerline{\psfig{figure=fig/radial.ps,height=4.5in}}\end{figure}$

Next: 8.1.3.4 Surface curvatures and Up: 8.1.3 Second-order interrogation methods Previous: 8.1.3.2 Zero curvature points Contents Index

December 2009