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8.1.3.2 Zero curvature points

The signed curvature formula for a planar parametric curve is given in (2.25). Due to the regularity condition, a necessary condition to determine inflection points is
$\displaystyle \dot{x}(t)\ddot{y}(t) - \dot{y}(t)\ddot{x}(t) = 0, \quad \quad t\in[t_1,t_2]\;.$     (8.8)

An inflection point on a planar curve is shown in Fig. 8.7, marked by $ \times $ .

The curvature $ \kappa (t)$ of a space curve is given in (2.26). The formula can be expressed as

$\displaystyle \kappa (t) = \frac{\sqrt{(\dot{x}\ddot{y} - \dot{y}\ddot{x})^2 +
...
...x} - \dot{x}\ddot{z})^2}}{(\dot{x}^2 + \dot{y}^2 + \dot{z}^2)^{\frac{3}{2}}}\;.$     (8.9)

Since we are assuming a regular curve, a condition to determine a point on a curve, where the curvature $ \kappa (t)$ vanishes, is [57]
$\displaystyle K_{0}(t)   {\bf\equiv}   (\dot{x}\ddot{y} - \dot{y}\ddot{x})^...
...{y})^2 + (\dot{z}\ddot{x} - \dot{x}\ddot{z})^2
=0, \quad \quad t\in[t_1,t_2]\;,$     (8.10)

or
$\displaystyle \dot{x}\ddot{y} - \dot{y}\ddot{x}=\dot{y}\ddot{z} - \dot{z}\ddot{y}= \dot{z}\ddot{x} - \dot{x}\ddot{z}=0, \quad \quad t\in[t_1,t_2]\;.$     (8.11)

Curvature vanishing points on a space curve $ {\bf r}(t)$ are shown in Fig. 8.8, marked by $ \times $ .


next up previous contents index
Next: 8.1.3.3 Radial curves Up: 8.1.3 Second-order interrogation methods Previous: 8.1.3.1 Curvature plots   Contents   Index
December 2009