8.1.5.2 Stationary points of total curvature

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8.1.5.2 Stationary points of total curvature

We now consider the notion of the Darboux vector $\Omega$ defined by [206],

$\displaystyle \mbox{\boldmath$\Omega$}$ $\displaystyle (t)=\tau (t){\bf t}(t) + \kappa (t) {\bf b}(t)\;,$

(8.32)

where ${\bf t}(t)$ and ${\bf b}(t)$ are the unit tangent and binormal vectors, respectively. The Darboux vector turns out to be a rotation vector of the Frenet frame while moving along the curve and therefore, its Euclidean norm $\vert$ $\Omega$ $(t)\vert$ indicates the angular speed $\omega (t)$ of the moving local frame. The angular speed $\omega (t)$ is sometimes called total curvature of a curve and defined by

$\displaystyle \omega (t) = \sqrt{\kappa^{2}(t) + \tau^{2}(t)}\;.$

(8.33)

In a planar curve, $\omega (t)$ reduces to $\kappa (t)$ and the binormal vector becomes the axis of rotation. We notice that the total curvature $\omega (t)$ captures the coupled effect of both intrinsic features of a space curve, and hence, we may consider it as a criterion function for detecting a significant point on a space curve [57].

At a nonzero curvature point on a regular space curve, where the moving frame has its locally highest or lowest angular speed, satisfies the equation $\dot\omega (t) = 0$ , i.e.

$\displaystyle {K_{0}^{3}}(t)K_{1}(t) + {G_{1}^{4}}(t)T_{0}(t)T_{1}(t) =0,\quad\quad t\in[t_1,t_2]\;,$

(8.34)

where , , $G_{1}(t)$ , $T_{0}(t)$ and $T_{1}(t)$ are defined in (8.10), (8.22), (8.23), (8.19) and (8.30), respectively. Comparing each function we can roughly see each contribution of $\kappa (t)$ , $\dot\kappa(t)$ , $\tau (t)$ and $\dot\tau (t)$ to (8.34). For a special example, if $\dot\kappa(t)$ and one of $\tau (t)$ or $\dot\tau (t)$ vanish at some , (8.34) is also satisfied there. We note $K_{0}(t)$ and $G_{1}(t)$ are always positive at a nonzero curvature point. Three points in Fig. 8.8 satisfy (8.34). Two points, marked by 's, are located close to the points of curvature extrema, and the other point, marked by $\oplus$ , is located at the midpoint of the curve where $\dot\kappa(t)$ and $\tau (t)$ also vanish.

Finally, for the special case where $\kappa (t)$ , $\tau (t)$ and consequently $\omega (t)$ are constant, the curve is a circular helix.

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December 2009