8.1.4.2 Stationary points of curvature of planar and space curves

Modern CAD/CAM systems allow users to access specific application programs for performing several tasks, such as displaying objects on a graphic display, mass property calculation, finite element or boundary element meshing for analysis. These application programs often operate on piecewise linear approximation of the exact geometric definition. When a coarse approximation of good quality is required for 2-D and 3-D curves, stationary points of curvature play an important role in successful discretization [193,97,150,57].

Using (2.25), the first derivative of the curvature function of a planar curve is given by

Since we are assuming a regular curve, the necessary condition to have a local maximum or minimum of curvature is given by

Figure 8.7 shows the corresponding stationary points of curvature on , labeled by . Furthermore, comparing (8.8) and (8.21) we see both and of a regular planar curve vanish for some if and only if and , simultaneously [57].

Figure 8.7: Significant points on a planar Bézier curve of degree 5 (adapted from [57])

Similarly for a regular space curve, the necessary condition to have stationary points of curvature at nonzero curvature points is given by

where

and is defined in (8.10). Note that at points , the necessary condition (8.22) is automatically satisfied (see (8.11)), thus at those points is equivalent to

Three points in Fig. 8.8 satisfy (8.22). Two of them are marked by 's and one is marked by at the midpoint of the curve.

Figure 8.8: Significant points on a space curve and its projections (adapted from [57])

Once all the stationary points are identified, we may use the extrema theory of functions for a single variable given in Sect. 7.3.1 to classify stationary points of curvature.